| L(s) = 1 | + 0.179·2-s + 1.24·3-s − 1.96·4-s − 5-s + 0.224·6-s − 7-s − 0.712·8-s − 1.44·9-s − 0.179·10-s − 4.52·11-s − 2.45·12-s − 6.55·13-s − 0.179·14-s − 1.24·15-s + 3.80·16-s − 4.38·17-s − 0.259·18-s − 1.34·19-s + 1.96·20-s − 1.24·21-s − 0.812·22-s − 2.35·23-s − 0.888·24-s + 25-s − 1.17·26-s − 5.54·27-s + 1.96·28-s + ⋯ |
| L(s) = 1 | + 0.127·2-s + 0.720·3-s − 0.983·4-s − 0.447·5-s + 0.0914·6-s − 0.377·7-s − 0.251·8-s − 0.481·9-s − 0.0568·10-s − 1.36·11-s − 0.708·12-s − 1.81·13-s − 0.0480·14-s − 0.322·15-s + 0.951·16-s − 1.06·17-s − 0.0611·18-s − 0.309·19-s + 0.439·20-s − 0.272·21-s − 0.173·22-s − 0.490·23-s − 0.181·24-s + 0.200·25-s − 0.230·26-s − 1.06·27-s + 0.371·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.06003781239\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.06003781239\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 229 | \( 1 - T \) |
| good | 2 | \( 1 - 0.179T + 2T^{2} \) |
| 3 | \( 1 - 1.24T + 3T^{2} \) |
| 11 | \( 1 + 4.52T + 11T^{2} \) |
| 13 | \( 1 + 6.55T + 13T^{2} \) |
| 17 | \( 1 + 4.38T + 17T^{2} \) |
| 19 | \( 1 + 1.34T + 19T^{2} \) |
| 23 | \( 1 + 2.35T + 23T^{2} \) |
| 29 | \( 1 - 1.86T + 29T^{2} \) |
| 31 | \( 1 + 6.41T + 31T^{2} \) |
| 37 | \( 1 - 10.5T + 37T^{2} \) |
| 41 | \( 1 + 0.929T + 41T^{2} \) |
| 43 | \( 1 + 6.66T + 43T^{2} \) |
| 47 | \( 1 + 12.1T + 47T^{2} \) |
| 53 | \( 1 + 8.21T + 53T^{2} \) |
| 59 | \( 1 - 4.77T + 59T^{2} \) |
| 61 | \( 1 + 2.51T + 61T^{2} \) |
| 67 | \( 1 + 5.21T + 67T^{2} \) |
| 71 | \( 1 + 11.2T + 71T^{2} \) |
| 73 | \( 1 - 13.1T + 73T^{2} \) |
| 79 | \( 1 + 13.0T + 79T^{2} \) |
| 83 | \( 1 + 1.40T + 83T^{2} \) |
| 89 | \( 1 + 14.6T + 89T^{2} \) |
| 97 | \( 1 - 9.91T + 97T^{2} \) |
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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
−7.81108478820804881818971550824, −7.51151179696951982041175736525, −6.44683178003732035995956226033, −5.57687013911350163451913610003, −4.84573159013462842736775181119, −4.42205314915322284103281481084, −3.38761045434230178212255357128, −2.80127828843643476353581665928, −2.08112924273550229718432440319, −0.10536485137275720326144183150,
0.10536485137275720326144183150, 2.08112924273550229718432440319, 2.80127828843643476353581665928, 3.38761045434230178212255357128, 4.42205314915322284103281481084, 4.84573159013462842736775181119, 5.57687013911350163451913610003, 6.44683178003732035995956226033, 7.51151179696951982041175736525, 7.81108478820804881818971550824
