L(s) = 1 | − 2-s + 4-s − 7-s − 8-s − 3·9-s − 11-s + 2·13-s + 14-s + 16-s − 4·17-s + 3·18-s + 4·19-s + 22-s + 4·23-s − 2·26-s − 28-s + 8·29-s − 2·31-s − 32-s + 4·34-s − 3·36-s − 10·37-s − 4·38-s − 6·41-s − 8·43-s − 44-s − 4·46-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.377·7-s − 0.353·8-s − 9-s − 0.301·11-s + 0.554·13-s + 0.267·14-s + 1/4·16-s − 0.970·17-s + 0.707·18-s + 0.917·19-s + 0.213·22-s + 0.834·23-s − 0.392·26-s − 0.188·28-s + 1.48·29-s − 0.359·31-s − 0.176·32-s + 0.685·34-s − 1/2·36-s − 1.64·37-s − 0.648·38-s − 0.937·41-s − 1.21·43-s − 0.150·44-s − 0.589·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9418191742\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9418191742\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
good | 3 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.677969353654589688303668286183, −7.956366388447664592623544100867, −6.87150502480062053477587745627, −6.58315995574118824567068813694, −5.51929752553932746343995070520, −4.92227219014542390048543539594, −3.53262376010960815263030689069, −2.95931954632752677435415814597, −1.91628790568383108446455220304, −0.60701948405204259149768805109,
0.60701948405204259149768805109, 1.91628790568383108446455220304, 2.95931954632752677435415814597, 3.53262376010960815263030689069, 4.92227219014542390048543539594, 5.51929752553932746343995070520, 6.58315995574118824567068813694, 6.87150502480062053477587745627, 7.956366388447664592623544100867, 8.677969353654589688303668286183