| L(s) = 1 | + 2·3-s − 2·7-s + 3·9-s − 4·13-s − 2·19-s − 4·21-s + 8·23-s + 2·25-s + 4·27-s − 8·29-s + 8·31-s − 4·37-s − 8·39-s + 12·41-s + 12·47-s + 3·49-s − 4·57-s + 8·59-s + 12·61-s − 6·63-s + 16·67-s + 16·69-s + 20·71-s + 12·73-s + 4·75-s + 5·81-s − 4·83-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.755·7-s + 9-s − 1.10·13-s − 0.458·19-s − 0.872·21-s + 1.66·23-s + 2/5·25-s + 0.769·27-s − 1.48·29-s + 1.43·31-s − 0.657·37-s − 1.28·39-s + 1.87·41-s + 1.75·47-s + 3/7·49-s − 0.529·57-s + 1.04·59-s + 1.53·61-s − 0.755·63-s + 1.95·67-s + 1.92·69-s + 2.37·71-s + 1.40·73-s + 0.461·75-s + 5/9·81-s − 0.439·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10188864 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10188864 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.075523661\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.075523661\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| 19 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 + 8 T + 62 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 8 T + 30 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 12 T + 118 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 94 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 - 16 T + 150 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 20 T + 230 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 12 T + 134 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 + 4 T + 158 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 4 T - 10 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.864524504841022728967437062290, −8.606470007460948377155255754626, −8.039084497724506173573685647429, −7.80061737834651605205031952013, −7.40308930132317026689638289573, −6.98038707783429464300890634491, −6.71105320129048704224351528357, −6.50814041155517083203089500109, −5.67026037850230400978382600014, −5.50527706997139096017408984842, −4.88320725999633880832106929064, −4.66686104195702559700868584536, −3.96200976144986388984838889691, −3.69838555790991062202755578297, −3.36329999769675379576435633807, −2.60780299944406925503146887254, −2.39255768519929206285759200004, −2.19726903968925207686209662830, −1.06469478135068637574654997320, −0.66285686493655192158641314752,
0.66285686493655192158641314752, 1.06469478135068637574654997320, 2.19726903968925207686209662830, 2.39255768519929206285759200004, 2.60780299944406925503146887254, 3.36329999769675379576435633807, 3.69838555790991062202755578297, 3.96200976144986388984838889691, 4.66686104195702559700868584536, 4.88320725999633880832106929064, 5.50527706997139096017408984842, 5.67026037850230400978382600014, 6.50814041155517083203089500109, 6.71105320129048704224351528357, 6.98038707783429464300890634491, 7.40308930132317026689638289573, 7.80061737834651605205031952013, 8.039084497724506173573685647429, 8.606470007460948377155255754626, 8.864524504841022728967437062290
