| L(s) = 1 | + 2·2-s − 4-s − 8·8-s − 2·9-s + 4·13-s − 7·16-s + 17-s − 4·18-s + 25-s + 8·26-s + 14·32-s + 2·34-s + 2·36-s + 8·43-s + 24·47-s − 10·49-s + 2·50-s − 4·52-s − 20·53-s + 16·59-s + 35·64-s + 16·67-s − 68-s + 16·72-s − 5·81-s + 8·83-s + 16·86-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1/2·4-s − 2.82·8-s − 2/3·9-s + 1.10·13-s − 7/4·16-s + 0.242·17-s − 0.942·18-s + 1/5·25-s + 1.56·26-s + 2.47·32-s + 0.342·34-s + 1/3·36-s + 1.21·43-s + 3.50·47-s − 1.42·49-s + 0.282·50-s − 0.554·52-s − 2.74·53-s + 2.08·59-s + 35/8·64-s + 1.95·67-s − 0.121·68-s + 1.88·72-s − 5/9·81-s + 0.878·83-s + 1.72·86-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 122825 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 122825 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.895215611\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.895215611\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 17 | $C_1$ | \( 1 - T \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−9.342260097121774077335621975199, −8.918128413786971033454816799413, −8.533766098435281827682513745391, −8.111471268075411853144192434707, −7.46613087643891033346359456413, −6.59958491014215133179348317151, −6.07635798451292745708773275520, −5.83398257972728633601666952176, −5.19219295929792253516288554932, −4.83415825893670616601536879622, −4.03297779911247219461657332614, −3.76631681926042089575770915559, −3.13757525842324917581154074415, −2.44164538755139501361428105925, −0.803480391876561932342091669982,
0.803480391876561932342091669982, 2.44164538755139501361428105925, 3.13757525842324917581154074415, 3.76631681926042089575770915559, 4.03297779911247219461657332614, 4.83415825893670616601536879622, 5.19219295929792253516288554932, 5.83398257972728633601666952176, 6.07635798451292745708773275520, 6.59958491014215133179348317151, 7.46613087643891033346359456413, 8.111471268075411853144192434707, 8.533766098435281827682513745391, 8.918128413786971033454816799413, 9.342260097121774077335621975199
