L(s) = 1 | + 2.07·2-s − 2.90·3-s + 2.30·4-s − 5-s − 6.01·6-s + 0.629·8-s + 5.41·9-s − 2.07·10-s + 2.88·11-s − 6.68·12-s − 1.20·13-s + 2.90·15-s − 3.30·16-s − 0.968·17-s + 11.2·18-s − 1.97·19-s − 2.30·20-s + 5.99·22-s + 23-s − 1.82·24-s + 25-s − 2.49·26-s − 7.01·27-s + 6.16·29-s + 6.01·30-s + 3.18·31-s − 8.10·32-s + ⋯ |
L(s) = 1 | + 1.46·2-s − 1.67·3-s + 1.15·4-s − 0.447·5-s − 2.45·6-s + 0.222·8-s + 1.80·9-s − 0.656·10-s + 0.871·11-s − 1.92·12-s − 0.334·13-s + 0.749·15-s − 0.825·16-s − 0.234·17-s + 2.64·18-s − 0.453·19-s − 0.515·20-s + 1.27·22-s + 0.208·23-s − 0.372·24-s + 0.200·25-s − 0.489·26-s − 1.35·27-s + 1.14·29-s + 1.09·30-s + 0.571·31-s − 1.43·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5635 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5635 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 23 | \( 1 - T \) |
good | 2 | \( 1 - 2.07T + 2T^{2} \) |
| 3 | \( 1 + 2.90T + 3T^{2} \) |
| 11 | \( 1 - 2.88T + 11T^{2} \) |
| 13 | \( 1 + 1.20T + 13T^{2} \) |
| 17 | \( 1 + 0.968T + 17T^{2} \) |
| 19 | \( 1 + 1.97T + 19T^{2} \) |
| 29 | \( 1 - 6.16T + 29T^{2} \) |
| 31 | \( 1 - 3.18T + 31T^{2} \) |
| 37 | \( 1 + 8.73T + 37T^{2} \) |
| 41 | \( 1 - 8.88T + 41T^{2} \) |
| 43 | \( 1 + 3.08T + 43T^{2} \) |
| 47 | \( 1 + 1.20T + 47T^{2} \) |
| 53 | \( 1 - 4.90T + 53T^{2} \) |
| 59 | \( 1 - 9.08T + 59T^{2} \) |
| 61 | \( 1 + 7.20T + 61T^{2} \) |
| 67 | \( 1 + 12.9T + 67T^{2} \) |
| 71 | \( 1 - 6.52T + 71T^{2} \) |
| 73 | \( 1 + 7.84T + 73T^{2} \) |
| 79 | \( 1 + 8.05T + 79T^{2} \) |
| 83 | \( 1 - 6.40T + 83T^{2} \) |
| 89 | \( 1 + 4.40T + 89T^{2} \) |
| 97 | \( 1 + 18.9T + 97T^{2} \) |
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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
−7.17719582756009530878311842744, −6.77212373802615400246099184191, −6.15687764496316934794261971314, −5.56951533716865911955882314897, −4.77389332089907897134616253825, −4.40322961319858853458611369704, −3.66639460577327779811606357226, −2.59813236483580470135245620640, −1.26581576647404564499338118911, 0,
1.26581576647404564499338118911, 2.59813236483580470135245620640, 3.66639460577327779811606357226, 4.40322961319858853458611369704, 4.77389332089907897134616253825, 5.56951533716865911955882314897, 6.15687764496316934794261971314, 6.77212373802615400246099184191, 7.17719582756009530878311842744