| L(s) = 1 | − 2-s + 4-s + 2·5-s − 8-s + 4·9-s − 2·10-s − 10·13-s + 16-s − 9·17-s − 4·18-s + 2·20-s − 25-s + 10·26-s + 14·29-s − 32-s + 9·34-s + 4·36-s + 2·37-s − 2·40-s − 10·41-s + 8·45-s + 9·49-s + 50-s − 10·52-s + 3·53-s − 14·58-s − 2·61-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.894·5-s − 0.353·8-s + 4/3·9-s − 0.632·10-s − 2.77·13-s + 1/4·16-s − 2.18·17-s − 0.942·18-s + 0.447·20-s − 1/5·25-s + 1.96·26-s + 2.59·29-s − 0.176·32-s + 1.54·34-s + 2/3·36-s + 0.328·37-s − 0.316·40-s − 1.56·41-s + 1.19·45-s + 9/7·49-s + 0.141·50-s − 1.38·52-s + 0.412·53-s − 1.83·58-s − 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1095200 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1095200 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.274105346\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.274105346\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 5 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 37 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 9 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 40 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 65 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 19 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 115 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 67 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 110 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
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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
−8.164761747383473998381471928086, −7.40481189007537705096929401020, −7.22044244614451391727593390327, −6.93650244814744888304505763088, −6.39963608045658517721263827746, −6.14861269786800413859638470365, −5.24010309071359340177032774287, −4.91651101580675004927187432110, −4.49872248887940796538835172341, −4.16938916346493811142842243048, −3.11210287513621025276935592771, −2.45436370347105642021038686479, −2.23581363610434537936476574962, −1.64536445355826177184284895642, −0.55320945815982753639827378470,
0.55320945815982753639827378470, 1.64536445355826177184284895642, 2.23581363610434537936476574962, 2.45436370347105642021038686479, 3.11210287513621025276935592771, 4.16938916346493811142842243048, 4.49872248887940796538835172341, 4.91651101580675004927187432110, 5.24010309071359340177032774287, 6.14861269786800413859638470365, 6.39963608045658517721263827746, 6.93650244814744888304505763088, 7.22044244614451391727593390327, 7.40481189007537705096929401020, 8.164761747383473998381471928086
