| L(s) = 1 | − 96·7-s + 404·9-s + 200·17-s − 2.33e3·23-s + 7.02e3·25-s + 1.29e4·31-s − 4.56e3·41-s + 5.47e4·47-s − 2.40e4·49-s − 3.87e4·63-s − 2.06e5·71-s + 3.99e4·73-s + 2.47e5·79-s + 7.90e4·81-s − 8.46e4·89-s − 9.95e4·97-s − 1.80e5·103-s + 3.03e5·113-s − 1.92e4·119-s + 2.96e5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 8.08e4·153-s + ⋯ |
| L(s) = 1 | − 0.740·7-s + 1.66·9-s + 0.167·17-s − 0.920·23-s + 2.24·25-s + 2.41·31-s − 0.424·41-s + 3.61·47-s − 1.43·49-s − 1.23·63-s − 4.86·71-s + 0.877·73-s + 4.46·79-s + 1.33·81-s − 1.13·89-s − 1.07·97-s − 1.67·103-s + 2.23·113-s − 0.124·119-s + 1.84·121-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + 3.69e−6·149-s + 3.56e−6·151-s + 0.279·153-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1048576 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1048576 ^{s/2} \, \Gamma_{\C}(s+5/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.278749248\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.278749248\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| good | 3 | $C_2^2 \wr C_2$ | \( 1 - 404 T^{2} + 9350 p^{2} T^{4} - 404 p^{10} T^{6} + p^{20} T^{8} \) |
| 5 | $C_2^2 \wr C_2$ | \( 1 - 7028 T^{2} + 24404246 T^{4} - 7028 p^{10} T^{6} + p^{20} T^{8} \) |
| 7 | $D_{4}$ | \( ( 1 + 48 T + 15502 T^{2} + 48 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 11 | $C_2^2 \wr C_2$ | \( 1 - 296436 T^{2} + 49128544726 T^{4} - 296436 p^{10} T^{6} + p^{20} T^{8} \) |
| 13 | $C_2^2 \wr C_2$ | \( 1 - 894228 T^{2} + 396323515894 T^{4} - 894228 p^{10} T^{6} + p^{20} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 - 100 T + 2767462 T^{2} - 100 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 19 | $C_2^2 \wr C_2$ | \( 1 - 6794580 T^{2} + 21506967947254 T^{4} - 6794580 p^{10} T^{6} + p^{20} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 + 1168 T + 10055470 T^{2} + 1168 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 29 | $C_2^2 \wr C_2$ | \( 1 - 31255380 T^{2} + 976386653995702 T^{4} - 31255380 p^{10} T^{6} + p^{20} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 - 6464 T + 65013054 T^{2} - 6464 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 37 | $C_2^2 \wr C_2$ | \( 1 - 241262580 T^{2} + 24149916431784598 T^{4} - 241262580 p^{10} T^{6} + p^{20} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 2284 T + 146603254 T^{2} + 2284 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 43 | $C_2^2 \wr C_2$ | \( 1 - 10845276 p T^{2} + 96250708269010006 T^{4} - 10845276 p^{11} T^{6} + p^{20} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 - 27360 T + 591338206 T^{2} - 27360 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 53 | $C_2^2 \wr C_2$ | \( 1 - 1039152180 T^{2} + 595616955270391126 T^{4} - 1039152180 p^{10} T^{6} + p^{20} T^{8} \) |
| 59 | $C_2^2 \wr C_2$ | \( 1 - 1537424180 T^{2} + 1399694789142612374 T^{4} - 1537424180 p^{10} T^{6} + p^{20} T^{8} \) |
| 61 | $C_2^2 \wr C_2$ | \( 1 + 741098540 T^{2} + 1478044222094100534 T^{4} + 741098540 p^{10} T^{6} + p^{20} T^{8} \) |
| 67 | $C_2^2 \wr C_2$ | \( 1 - 1366835860 T^{2} + 3274116308996825526 T^{4} - 1366835860 p^{10} T^{6} + p^{20} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 103344 T + 6217736974 T^{2} + 103344 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 73 | $D_{4}$ | \( ( 1 - 19988 T + 2541602870 T^{2} - 19988 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 - 123936 T + 9855929374 T^{2} - 123936 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 83 | $C_2^2 \wr C_2$ | \( 1 - 10047855188 T^{2} + 55381071937674414326 T^{4} - 10047855188 p^{10} T^{6} + p^{20} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 42316 T + 4292401174 T^{2} + 42316 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 + 49788 T + 16391371462 T^{2} + 49788 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.95292660248357466358564265180, −10.96726838034995488869046341602, −10.87926642551199050227714688080, −10.54519985194797584919174342405, −10.17538672952676627118221845070, −9.810033824693913932089497285715, −9.751953214143535884346290870259, −9.016836970479024740168869902060, −8.965111317482679452070749772155, −8.406398455997316278390177555674, −7.920009510989547500030322119696, −7.66147134868770394633779944625, −7.07546570286264314897418744525, −6.71610573358325930208941729867, −6.66074353373509311101019341913, −5.92438027993361445072692928814, −5.62557648523865038144034692479, −4.79240772549305265314692755971, −4.45266756426103196325240542233, −4.20531429105249763560298576626, −3.32846712608163654962922153318, −2.91646349606784807646718890878, −2.13819946288637623989555023354, −1.21401113985390081670342059595, −0.67708065842039853067032152930,
0.67708065842039853067032152930, 1.21401113985390081670342059595, 2.13819946288637623989555023354, 2.91646349606784807646718890878, 3.32846712608163654962922153318, 4.20531429105249763560298576626, 4.45266756426103196325240542233, 4.79240772549305265314692755971, 5.62557648523865038144034692479, 5.92438027993361445072692928814, 6.66074353373509311101019341913, 6.71610573358325930208941729867, 7.07546570286264314897418744525, 7.66147134868770394633779944625, 7.920009510989547500030322119696, 8.406398455997316278390177555674, 8.965111317482679452070749772155, 9.016836970479024740168869902060, 9.751953214143535884346290870259, 9.810033824693913932089497285715, 10.17538672952676627118221845070, 10.54519985194797584919174342405, 10.87926642551199050227714688080, 10.96726838034995488869046341602, 11.95292660248357466358564265180
