L(s) = 1 | + 0.471·3-s + 0.567·7-s − 2.77·9-s − 4.37·11-s + 0.165·13-s + 7.94·17-s − 19-s + 0.267·21-s + 3.87·23-s − 2.72·27-s + 3.53·29-s − 3.20·31-s − 2.06·33-s − 10.1·37-s + 0.0779·39-s − 5.97·41-s + 12.0·43-s − 5.46·47-s − 6.67·49-s + 3.74·51-s + 2.00·53-s − 0.471·57-s − 8.32·59-s + 11.8·61-s − 1.57·63-s + 8.79·67-s + 1.82·69-s + ⋯ |
L(s) = 1 | + 0.271·3-s + 0.214·7-s − 0.926·9-s − 1.32·11-s + 0.0458·13-s + 1.92·17-s − 0.229·19-s + 0.0583·21-s + 0.807·23-s − 0.523·27-s + 0.655·29-s − 0.575·31-s − 0.358·33-s − 1.67·37-s + 0.0124·39-s − 0.932·41-s + 1.84·43-s − 0.796·47-s − 0.954·49-s + 0.524·51-s + 0.275·53-s − 0.0623·57-s − 1.08·59-s + 1.51·61-s − 0.198·63-s + 1.07·67-s + 0.219·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 3 | \( 1 - 0.471T + 3T^{2} \) |
| 7 | \( 1 - 0.567T + 7T^{2} \) |
| 11 | \( 1 + 4.37T + 11T^{2} \) |
| 13 | \( 1 - 0.165T + 13T^{2} \) |
| 17 | \( 1 - 7.94T + 17T^{2} \) |
| 23 | \( 1 - 3.87T + 23T^{2} \) |
| 29 | \( 1 - 3.53T + 29T^{2} \) |
| 31 | \( 1 + 3.20T + 31T^{2} \) |
| 37 | \( 1 + 10.1T + 37T^{2} \) |
| 41 | \( 1 + 5.97T + 41T^{2} \) |
| 43 | \( 1 - 12.0T + 43T^{2} \) |
| 47 | \( 1 + 5.46T + 47T^{2} \) |
| 53 | \( 1 - 2.00T + 53T^{2} \) |
| 59 | \( 1 + 8.32T + 59T^{2} \) |
| 61 | \( 1 - 11.8T + 61T^{2} \) |
| 67 | \( 1 - 8.79T + 67T^{2} \) |
| 71 | \( 1 + 0.720T + 71T^{2} \) |
| 73 | \( 1 + 4.54T + 73T^{2} \) |
| 79 | \( 1 + 11.7T + 79T^{2} \) |
| 83 | \( 1 + 6.72T + 83T^{2} \) |
| 89 | \( 1 - 8.11T + 89T^{2} \) |
| 97 | \( 1 - 13.6T + 97T^{2} \) |
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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
−7.68909317326917668645600076380, −6.97047579900342943959762309364, −5.99071715937105998111442160912, −5.31167279611936087951353443151, −4.99857297745945653659574786172, −3.70110372760700890770195139898, −3.09205691665947725491564995582, −2.42255912755887699568598329853, −1.28045491613866060849014307113, 0,
1.28045491613866060849014307113, 2.42255912755887699568598329853, 3.09205691665947725491564995582, 3.70110372760700890770195139898, 4.99857297745945653659574786172, 5.31167279611936087951353443151, 5.99071715937105998111442160912, 6.97047579900342943959762309364, 7.68909317326917668645600076380