| L(s) = 1 | − 2·2-s − 3-s + 2·4-s + 2·6-s − 2·7-s + 9-s − 2·12-s + 4·14-s − 4·16-s − 2·18-s − 4·19-s + 2·21-s − 25-s − 27-s − 4·28-s + 2·29-s + 8·32-s + 2·36-s + 8·38-s − 4·41-s − 4·42-s − 8·43-s + 4·48-s − 7·49-s + 2·50-s + 16·53-s + 2·54-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 0.577·3-s + 4-s + 0.816·6-s − 0.755·7-s + 1/3·9-s − 0.577·12-s + 1.06·14-s − 16-s − 0.471·18-s − 0.917·19-s + 0.436·21-s − 1/5·25-s − 0.192·27-s − 0.755·28-s + 0.371·29-s + 1.41·32-s + 1/3·36-s + 1.29·38-s − 0.624·41-s − 0.617·42-s − 1.21·43-s + 0.577·48-s − 49-s + 0.282·50-s + 2.19·53-s + 0.272·54-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 623808 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 623808 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3227865330\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3227865330\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 3 | $C_1$ | \( 1 + T \) |
| 19 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 7 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 29 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 11 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + 13 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 110 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 42 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
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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
−8.473911040505740234049264093490, −8.097703934664376855933869112895, −7.47631775423251166858572716196, −7.05301532131073450262486976017, −6.78028970322457743586764739367, −6.20866742393400467343223426123, −5.90554248252964034027323531150, −5.17641483466606412124476404625, −4.64876894290830996668792492313, −4.16945939289826073736551708354, −3.47977466697865119481167161641, −2.81077415533186876568916794423, −2.05320169733718604030558912314, −1.40981953234231144543937581253, −0.38788195122073911154862407440,
0.38788195122073911154862407440, 1.40981953234231144543937581253, 2.05320169733718604030558912314, 2.81077415533186876568916794423, 3.47977466697865119481167161641, 4.16945939289826073736551708354, 4.64876894290830996668792492313, 5.17641483466606412124476404625, 5.90554248252964034027323531150, 6.20866742393400467343223426123, 6.78028970322457743586764739367, 7.05301532131073450262486976017, 7.47631775423251166858572716196, 8.097703934664376855933869112895, 8.473911040505740234049264093490
