| L(s) = 1 | + 3.13·3-s − 3.40·5-s − 1.43·7-s + 6.84·9-s − 6.08·11-s + 2.68·13-s − 10.6·15-s − 4.18·17-s + 4.13·19-s − 4.49·21-s − 6.04·23-s + 6.61·25-s + 12.0·27-s + 5.24·29-s + 2.46·31-s − 19.0·33-s + 4.87·35-s + 5.72·37-s + 8.41·39-s + 7.25·41-s + 9.99·43-s − 23.3·45-s + 2.57·47-s − 4.94·49-s − 13.1·51-s − 5.63·53-s + 20.7·55-s + ⋯ |
| L(s) = 1 | + 1.81·3-s − 1.52·5-s − 0.541·7-s + 2.28·9-s − 1.83·11-s + 0.743·13-s − 2.76·15-s − 1.01·17-s + 0.948·19-s − 0.980·21-s − 1.25·23-s + 1.32·25-s + 2.32·27-s + 0.973·29-s + 0.443·31-s − 3.32·33-s + 0.824·35-s + 0.941·37-s + 1.34·39-s + 1.13·41-s + 1.52·43-s − 3.47·45-s + 0.375·47-s − 0.707·49-s − 1.84·51-s − 0.774·53-s + 2.79·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.344025160\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.344025160\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 751 | \( 1 - T \) |
| good | 3 | \( 1 - 3.13T + 3T^{2} \) |
| 5 | \( 1 + 3.40T + 5T^{2} \) |
| 7 | \( 1 + 1.43T + 7T^{2} \) |
| 11 | \( 1 + 6.08T + 11T^{2} \) |
| 13 | \( 1 - 2.68T + 13T^{2} \) |
| 17 | \( 1 + 4.18T + 17T^{2} \) |
| 19 | \( 1 - 4.13T + 19T^{2} \) |
| 23 | \( 1 + 6.04T + 23T^{2} \) |
| 29 | \( 1 - 5.24T + 29T^{2} \) |
| 31 | \( 1 - 2.46T + 31T^{2} \) |
| 37 | \( 1 - 5.72T + 37T^{2} \) |
| 41 | \( 1 - 7.25T + 41T^{2} \) |
| 43 | \( 1 - 9.99T + 43T^{2} \) |
| 47 | \( 1 - 2.57T + 47T^{2} \) |
| 53 | \( 1 + 5.63T + 53T^{2} \) |
| 59 | \( 1 - 3.82T + 59T^{2} \) |
| 61 | \( 1 + 3.66T + 61T^{2} \) |
| 67 | \( 1 - 13.8T + 67T^{2} \) |
| 71 | \( 1 - 6.02T + 71T^{2} \) |
| 73 | \( 1 - 12.9T + 73T^{2} \) |
| 79 | \( 1 - 3.40T + 79T^{2} \) |
| 83 | \( 1 - 8.72T + 83T^{2} \) |
| 89 | \( 1 + 11.9T + 89T^{2} \) |
| 97 | \( 1 + 4.94T + 97T^{2} \) |
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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
−8.205196335276114072903678326435, −7.67098735794228809476034018400, −7.06758592891557806987953250725, −6.09834904210276431071548182081, −4.84371682574881988372037927750, −4.14437660080238430290100313562, −3.57085469209232986633616704436, −2.81839467156979958732488188896, −2.31289848672053915501088497511, −0.71043641977047359559155548249,
0.71043641977047359559155548249, 2.31289848672053915501088497511, 2.81839467156979958732488188896, 3.57085469209232986633616704436, 4.14437660080238430290100313562, 4.84371682574881988372037927750, 6.09834904210276431071548182081, 7.06758592891557806987953250725, 7.67098735794228809476034018400, 8.205196335276114072903678326435
