L(s) = 1 | − 2-s + 3-s + 4-s + 0.436·5-s − 6-s − 4.48·7-s − 8-s + 9-s − 0.436·10-s − 1.73·11-s + 12-s + 5.08·13-s + 4.48·14-s + 0.436·15-s + 16-s − 0.656·17-s − 18-s − 19-s + 0.436·20-s − 4.48·21-s + 1.73·22-s + 7.45·23-s − 24-s − 4.80·25-s − 5.08·26-s + 27-s − 4.48·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.195·5-s − 0.408·6-s − 1.69·7-s − 0.353·8-s + 0.333·9-s − 0.138·10-s − 0.523·11-s + 0.288·12-s + 1.41·13-s + 1.19·14-s + 0.112·15-s + 0.250·16-s − 0.159·17-s − 0.235·18-s − 0.229·19-s + 0.0975·20-s − 0.978·21-s + 0.369·22-s + 1.55·23-s − 0.204·24-s − 0.961·25-s − 0.998·26-s + 0.192·27-s − 0.847·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.319526445\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.319526445\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| 53 | \( 1 + T \) |
good | 5 | \( 1 - 0.436T + 5T^{2} \) |
| 7 | \( 1 + 4.48T + 7T^{2} \) |
| 11 | \( 1 + 1.73T + 11T^{2} \) |
| 13 | \( 1 - 5.08T + 13T^{2} \) |
| 17 | \( 1 + 0.656T + 17T^{2} \) |
| 23 | \( 1 - 7.45T + 23T^{2} \) |
| 29 | \( 1 + 2.33T + 29T^{2} \) |
| 31 | \( 1 - 2.69T + 31T^{2} \) |
| 37 | \( 1 + 6.70T + 37T^{2} \) |
| 41 | \( 1 + 2.17T + 41T^{2} \) |
| 43 | \( 1 - 7.23T + 43T^{2} \) |
| 47 | \( 1 - 0.322T + 47T^{2} \) |
| 59 | \( 1 - 0.584T + 59T^{2} \) |
| 61 | \( 1 - 3.23T + 61T^{2} \) |
| 67 | \( 1 - 1.52T + 67T^{2} \) |
| 71 | \( 1 + 7.62T + 71T^{2} \) |
| 73 | \( 1 + 8.48T + 73T^{2} \) |
| 79 | \( 1 - 6.69T + 79T^{2} \) |
| 83 | \( 1 + 4.71T + 83T^{2} \) |
| 89 | \( 1 - 8.51T + 89T^{2} \) |
| 97 | \( 1 + 4.39T + 97T^{2} \) |
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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
−8.244151980278854257880137481110, −7.34904917627634690491420851011, −6.77752152998899483279283178695, −6.13450780894897277804179381694, −5.48013773583491555018604335111, −4.16439603405523283874404439808, −3.34589602675289400209473218060, −2.87378959189165420735242308061, −1.82719573087102497607771249493, −0.64056609324540604316253402698,
0.64056609324540604316253402698, 1.82719573087102497607771249493, 2.87378959189165420735242308061, 3.34589602675289400209473218060, 4.16439603405523283874404439808, 5.48013773583491555018604335111, 6.13450780894897277804179381694, 6.77752152998899483279283178695, 7.34904917627634690491420851011, 8.244151980278854257880137481110