| L(s) = 1 | − 2·5-s + 2·9-s + 8·11-s + 2·19-s − 25-s − 4·29-s − 16·31-s − 12·41-s − 4·45-s + 10·49-s − 16·55-s + 8·59-s + 12·61-s + 32·71-s − 16·79-s − 5·81-s + 20·89-s − 4·95-s + 16·99-s + 12·101-s − 4·109-s + 26·121-s + 12·125-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 2/3·9-s + 2.41·11-s + 0.458·19-s − 1/5·25-s − 0.742·29-s − 2.87·31-s − 1.87·41-s − 0.596·45-s + 10/7·49-s − 2.15·55-s + 1.04·59-s + 1.53·61-s + 3.79·71-s − 1.80·79-s − 5/9·81-s + 2.11·89-s − 0.410·95-s + 1.60·99-s + 1.19·101-s − 0.383·109-s + 2.36·121-s + 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 577600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 577600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.858290231\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.858290231\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 66 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 178 T^{2} + p^{2} T^{4} \) |
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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
−10.56007352820848211613586236695, −10.04420388066293746691472481252, −9.628065229770888312520433324806, −9.213329959868014563006953866881, −8.970851880586803498031509548749, −8.448225450643892751223301091170, −7.999485720007935767725731141832, −7.33220318395874269312754255883, −7.10936673946389006911433229287, −6.80304340661402684986409165635, −6.29154039178868606588097619726, −5.52623443696946072614658338147, −5.30556461903066781720812907295, −4.47420773586756922187031318957, −3.91386723811937542793690976495, −3.69260954042474352820617089829, −3.43425347629232948359136292914, −2.09508991772656035821486912281, −1.67296844234517499522044877707, −0.73782536169861143559388146893,
0.73782536169861143559388146893, 1.67296844234517499522044877707, 2.09508991772656035821486912281, 3.43425347629232948359136292914, 3.69260954042474352820617089829, 3.91386723811937542793690976495, 4.47420773586756922187031318957, 5.30556461903066781720812907295, 5.52623443696946072614658338147, 6.29154039178868606588097619726, 6.80304340661402684986409165635, 7.10936673946389006911433229287, 7.33220318395874269312754255883, 7.999485720007935767725731141832, 8.448225450643892751223301091170, 8.970851880586803498031509548749, 9.213329959868014563006953866881, 9.628065229770888312520433324806, 10.04420388066293746691472481252, 10.56007352820848211613586236695
