L(s) = 1 | − 2-s + 4-s − 0.732·7-s − 8-s − 3.73·11-s + 2.73·13-s + 0.732·14-s + 16-s − 6.19·17-s − 19-s + 3.73·22-s + 5.92·23-s − 2.73·26-s − 0.732·28-s + 1.73·29-s − 2.46·31-s − 32-s + 6.19·34-s + 2·37-s + 38-s + 2.92·41-s + 8.19·43-s − 3.73·44-s − 5.92·46-s + 3.46·47-s − 6.46·49-s + 2.73·52-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.276·7-s − 0.353·8-s − 1.12·11-s + 0.757·13-s + 0.195·14-s + 0.250·16-s − 1.50·17-s − 0.229·19-s + 0.795·22-s + 1.23·23-s − 0.535·26-s − 0.138·28-s + 0.321·29-s − 0.442·31-s − 0.176·32-s + 1.06·34-s + 0.328·37-s + 0.162·38-s + 0.457·41-s + 1.24·43-s − 0.562·44-s − 0.874·46-s + 0.505·47-s − 0.923·49-s + 0.378·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 7 | \( 1 + 0.732T + 7T^{2} \) |
| 11 | \( 1 + 3.73T + 11T^{2} \) |
| 13 | \( 1 - 2.73T + 13T^{2} \) |
| 17 | \( 1 + 6.19T + 17T^{2} \) |
| 23 | \( 1 - 5.92T + 23T^{2} \) |
| 29 | \( 1 - 1.73T + 29T^{2} \) |
| 31 | \( 1 + 2.46T + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 - 2.92T + 41T^{2} \) |
| 43 | \( 1 - 8.19T + 43T^{2} \) |
| 47 | \( 1 - 3.46T + 47T^{2} \) |
| 53 | \( 1 + 1.73T + 53T^{2} \) |
| 59 | \( 1 - 8.19T + 59T^{2} \) |
| 61 | \( 1 - 10.6T + 61T^{2} \) |
| 67 | \( 1 - 0.267T + 67T^{2} \) |
| 71 | \( 1 + 12.1T + 71T^{2} \) |
| 73 | \( 1 - 2.46T + 73T^{2} \) |
| 79 | \( 1 + 5.53T + 79T^{2} \) |
| 83 | \( 1 - 3.73T + 83T^{2} \) |
| 89 | \( 1 + 10.8T + 89T^{2} \) |
| 97 | \( 1 - 15.1T + 97T^{2} \) |
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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
−7.45951049788293498263211096513, −6.85887187534275611072641475834, −6.20841489097919530459648877439, −5.46955190622679789464938566091, −4.65671713790458360077584951458, −3.80527472715186443151093871115, −2.80462236326792919036270373649, −2.27033663724922039899855640776, −1.08569765668996990474122397801, 0,
1.08569765668996990474122397801, 2.27033663724922039899855640776, 2.80462236326792919036270373649, 3.80527472715186443151093871115, 4.65671713790458360077584951458, 5.46955190622679789464938566091, 6.20841489097919530459648877439, 6.85887187534275611072641475834, 7.45951049788293498263211096513