| L(s) = 1 | − 12·9-s − 6·11-s − 2·17-s + 4·19-s − 9·25-s + 12·41-s − 26·43-s − 5·49-s − 4·59-s + 4·67-s − 2·73-s + 90·81-s + 8·83-s + 40·89-s + 36·97-s + 72·99-s − 36·107-s + 20·113-s + 7·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 24·153-s + 157-s + ⋯ |
| L(s) = 1 | − 4·9-s − 1.80·11-s − 0.485·17-s + 0.917·19-s − 9/5·25-s + 1.87·41-s − 3.96·43-s − 5/7·49-s − 0.520·59-s + 0.488·67-s − 0.234·73-s + 10·81-s + 0.878·83-s + 4.23·89-s + 3.65·97-s + 7.23·99-s − 3.48·107-s + 1.88·113-s + 7/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 1.94·153-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4926701062\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4926701062\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 - T )^{4} \) |
| good | 3 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 5 | $C_2^3$ | \( 1 + 9 T^{2} + 56 T^{4} + 9 p^{2} T^{6} + p^{4} T^{8} \) |
| 7 | $C_2^3$ | \( 1 + 5 T^{2} - 24 T^{4} + 5 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 + 3 T + 10 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 + 8 T^{2} + 126 T^{4} + 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 + T + 20 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $D_4\times C_2$ | \( 1 + 72 T^{2} + 2750 T^{4} + 72 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 + 80 T^{2} + 3294 T^{4} + 80 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 56 T^{2} + 1470 T^{4} + 56 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 6 T + 34 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 + 13 T + 114 T^{2} + 13 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 + 165 T^{2} + 11096 T^{4} + 165 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $D_{4}$ | \( ( 1 + 2 T + 62 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 + 233 T^{2} + 21000 T^{4} + 233 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 - 2 T + 78 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 - 64 T^{2} + 9054 T^{4} - 64 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 + T + 132 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 + 128 T^{2} + 10878 T^{4} + 128 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 - 4 T - 58 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{4} \) |
| 97 | $D_{4}$ | \( ( 1 - 18 T + 218 T^{2} - 18 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−5.96487139318649494980979474388, −5.51438099941034511068897114941, −5.42271860480826455429214689125, −5.33624686491283918795399199029, −5.13058513653440722782966096546, −4.88850981608777760005677890564, −4.81432936212083945523418895992, −4.63943580040744344666674702517, −4.38910853117492039148551883241, −3.84872943114955712686036320777, −3.69476630574732800184472064293, −3.52463949670690475110474833056, −3.41079337621861222421675595322, −3.19591401357625292898022614057, −2.92901905890559488010402701338, −2.74645279872635798735674039570, −2.72115558296687771466792913603, −2.15812031552203834523131109397, −2.11406824056540918147584142238, −2.02070760864862168114678966504, −1.78547202763414793049203146516, −1.05740655751382970555992298126, −0.792884782958454741270338568494, −0.34667117801924924783124317176, −0.20515519765559077076645712452,
0.20515519765559077076645712452, 0.34667117801924924783124317176, 0.792884782958454741270338568494, 1.05740655751382970555992298126, 1.78547202763414793049203146516, 2.02070760864862168114678966504, 2.11406824056540918147584142238, 2.15812031552203834523131109397, 2.72115558296687771466792913603, 2.74645279872635798735674039570, 2.92901905890559488010402701338, 3.19591401357625292898022614057, 3.41079337621861222421675595322, 3.52463949670690475110474833056, 3.69476630574732800184472064293, 3.84872943114955712686036320777, 4.38910853117492039148551883241, 4.63943580040744344666674702517, 4.81432936212083945523418895992, 4.88850981608777760005677890564, 5.13058513653440722782966096546, 5.33624686491283918795399199029, 5.42271860480826455429214689125, 5.51438099941034511068897114941, 5.96487139318649494980979474388
