L(s) = 1 | + 2-s + 2·3-s + 4-s + 2·6-s − 7-s + 8-s + 9-s − 6·11-s + 2·12-s + 13-s − 14-s + 16-s + 3·17-s + 18-s − 4·19-s − 2·21-s − 6·22-s + 23-s + 2·24-s + 26-s − 4·27-s − 28-s − 6·29-s + 8·31-s + 32-s − 12·33-s + 3·34-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.15·3-s + 1/2·4-s + 0.816·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s − 1.80·11-s + 0.577·12-s + 0.277·13-s − 0.267·14-s + 1/4·16-s + 0.727·17-s + 0.235·18-s − 0.917·19-s − 0.436·21-s − 1.27·22-s + 0.208·23-s + 0.408·24-s + 0.196·26-s − 0.769·27-s − 0.188·28-s − 1.11·29-s + 1.43·31-s + 0.176·32-s − 2.08·33-s + 0.514·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 23 | \( 1 - T \) |
good | 3 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 11 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−7.59604280605560956175024427547, −6.84220158127032164843302897128, −6.00212874467869519919736116645, −5.31734092371358798587742391545, −4.66086393172301257530890512148, −3.62087777678666329916006498037, −3.15510815113127334526718651560, −2.50007576998651018891514162695, −1.72376764281066899262318416372, 0,
1.72376764281066899262318416372, 2.50007576998651018891514162695, 3.15510815113127334526718651560, 3.62087777678666329916006498037, 4.66086393172301257530890512148, 5.31734092371358798587742391545, 6.00212874467869519919736116645, 6.84220158127032164843302897128, 7.59604280605560956175024427547