| L(s) = 1 | − 2-s − 1.60·3-s + 4-s + 4.22·5-s + 1.60·6-s − 0.325·7-s − 8-s − 0.410·9-s − 4.22·10-s + 3.83·11-s − 1.60·12-s + 0.325·14-s − 6.80·15-s + 16-s + 2.40·17-s + 0.410·18-s − 19-s + 4.22·20-s + 0.524·21-s − 3.83·22-s + 2.11·23-s + 1.60·24-s + 12.8·25-s + 5.48·27-s − 0.325·28-s + 7.46·29-s + 6.80·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.929·3-s + 0.5·4-s + 1.88·5-s + 0.656·6-s − 0.123·7-s − 0.353·8-s − 0.136·9-s − 1.33·10-s + 1.15·11-s − 0.464·12-s + 0.0870·14-s − 1.75·15-s + 0.250·16-s + 0.583·17-s + 0.0966·18-s − 0.229·19-s + 0.944·20-s + 0.114·21-s − 0.816·22-s + 0.440·23-s + 0.328·24-s + 2.57·25-s + 1.05·27-s − 0.0615·28-s + 1.38·29-s + 1.24·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.689192010\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.689192010\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 13 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + 1.60T + 3T^{2} \) |
| 5 | \( 1 - 4.22T + 5T^{2} \) |
| 7 | \( 1 + 0.325T + 7T^{2} \) |
| 11 | \( 1 - 3.83T + 11T^{2} \) |
| 17 | \( 1 - 2.40T + 17T^{2} \) |
| 23 | \( 1 - 2.11T + 23T^{2} \) |
| 29 | \( 1 - 7.46T + 29T^{2} \) |
| 31 | \( 1 - 4.53T + 31T^{2} \) |
| 37 | \( 1 + 2.51T + 37T^{2} \) |
| 41 | \( 1 + 8.17T + 41T^{2} \) |
| 43 | \( 1 + 8.05T + 43T^{2} \) |
| 47 | \( 1 - 1.42T + 47T^{2} \) |
| 53 | \( 1 - 8.33T + 53T^{2} \) |
| 59 | \( 1 + 6.88T + 59T^{2} \) |
| 61 | \( 1 - 9.16T + 61T^{2} \) |
| 67 | \( 1 - 1.83T + 67T^{2} \) |
| 71 | \( 1 - 11.6T + 71T^{2} \) |
| 73 | \( 1 - 12.3T + 73T^{2} \) |
| 79 | \( 1 + 14.2T + 79T^{2} \) |
| 83 | \( 1 - 6.05T + 83T^{2} \) |
| 89 | \( 1 + 0.553T + 89T^{2} \) |
| 97 | \( 1 - 2.52T + 97T^{2} \) |
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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
−8.334793230417957940721272931459, −6.86136543969726190556856999098, −6.60719300544402962729645458235, −6.09444816746822918101729665892, −5.33417082579079338360741084062, −4.81251888944734320415255957767, −3.38042027231668998156609331536, −2.50504409084609303873193771514, −1.55176214367110650230804405893, −0.853383987332098862444282876612,
0.853383987332098862444282876612, 1.55176214367110650230804405893, 2.50504409084609303873193771514, 3.38042027231668998156609331536, 4.81251888944734320415255957767, 5.33417082579079338360741084062, 6.09444816746822918101729665892, 6.60719300544402962729645458235, 6.86136543969726190556856999098, 8.334793230417957940721272931459
