| L(s) = 1 | − 1.24·2-s − 2.95·3-s − 0.455·4-s + 2.31·5-s + 3.67·6-s − 2.50·7-s + 3.05·8-s + 5.75·9-s − 2.87·10-s + 1.34·12-s − 4.51·13-s + 3.10·14-s − 6.84·15-s − 2.88·16-s − 1.92·17-s − 7.15·18-s − 7.30·19-s − 1.05·20-s + 7.40·21-s + 4.71·23-s − 9.03·24-s + 0.355·25-s + 5.61·26-s − 8.16·27-s + 1.13·28-s − 5.82·29-s + 8.51·30-s + ⋯ |
| L(s) = 1 | − 0.878·2-s − 1.70·3-s − 0.227·4-s + 1.03·5-s + 1.50·6-s − 0.945·7-s + 1.07·8-s + 1.91·9-s − 0.909·10-s + 0.388·12-s − 1.25·13-s + 0.830·14-s − 1.76·15-s − 0.720·16-s − 0.467·17-s − 1.68·18-s − 1.67·19-s − 0.235·20-s + 1.61·21-s + 0.983·23-s − 1.84·24-s + 0.0710·25-s + 1.10·26-s − 1.57·27-s + 0.215·28-s − 1.08·29-s + 1.55·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7381 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.04649671055\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.04649671055\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 \) |
| 61 | \( 1 + T \) |
| good | 2 | \( 1 + 1.24T + 2T^{2} \) |
| 3 | \( 1 + 2.95T + 3T^{2} \) |
| 5 | \( 1 - 2.31T + 5T^{2} \) |
| 7 | \( 1 + 2.50T + 7T^{2} \) |
| 13 | \( 1 + 4.51T + 13T^{2} \) |
| 17 | \( 1 + 1.92T + 17T^{2} \) |
| 19 | \( 1 + 7.30T + 19T^{2} \) |
| 23 | \( 1 - 4.71T + 23T^{2} \) |
| 29 | \( 1 + 5.82T + 29T^{2} \) |
| 31 | \( 1 + 1.06T + 31T^{2} \) |
| 37 | \( 1 + 2.79T + 37T^{2} \) |
| 41 | \( 1 + 8.69T + 41T^{2} \) |
| 43 | \( 1 + 4.66T + 43T^{2} \) |
| 47 | \( 1 + 1.70T + 47T^{2} \) |
| 53 | \( 1 + 12.0T + 53T^{2} \) |
| 59 | \( 1 - 6.98T + 59T^{2} \) |
| 67 | \( 1 + 5.99T + 67T^{2} \) |
| 71 | \( 1 - 13.7T + 71T^{2} \) |
| 73 | \( 1 + 6.94T + 73T^{2} \) |
| 79 | \( 1 - 10.3T + 79T^{2} \) |
| 83 | \( 1 - 1.00T + 83T^{2} \) |
| 89 | \( 1 + 6.01T + 89T^{2} \) |
| 97 | \( 1 + 5.25T + 97T^{2} \) |
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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
−7.82847664296134455553612454817, −6.89108918762980174317558860073, −6.63633724103437021639139169504, −5.91233751094766590534746647454, −5.05006391140103875899961547771, −4.78273270660268253901681448363, −3.69555325035616898030464700203, −2.26334684907223245906474735663, −1.49946882997894152553934453989, −0.13855872000102442929644863074,
0.13855872000102442929644863074, 1.49946882997894152553934453989, 2.26334684907223245906474735663, 3.69555325035616898030464700203, 4.78273270660268253901681448363, 5.05006391140103875899961547771, 5.91233751094766590534746647454, 6.63633724103437021639139169504, 6.89108918762980174317558860073, 7.82847664296134455553612454817
