L(s) = 1 | + 2-s + 2.57·3-s + 4-s − 0.396·5-s + 2.57·6-s − 0.271·7-s + 8-s + 3.62·9-s − 0.396·10-s − 3.20·11-s + 2.57·12-s + 4.71·13-s − 0.271·14-s − 1.02·15-s + 16-s + 6.19·17-s + 3.62·18-s + 1.43·19-s − 0.396·20-s − 0.698·21-s − 3.20·22-s − 1.57·23-s + 2.57·24-s − 4.84·25-s + 4.71·26-s + 1.60·27-s − 0.271·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.48·3-s + 0.5·4-s − 0.177·5-s + 1.05·6-s − 0.102·7-s + 0.353·8-s + 1.20·9-s − 0.125·10-s − 0.966·11-s + 0.742·12-s + 1.30·13-s − 0.0725·14-s − 0.263·15-s + 0.250·16-s + 1.50·17-s + 0.853·18-s + 0.328·19-s − 0.0886·20-s − 0.152·21-s − 0.683·22-s − 0.328·23-s + 0.525·24-s − 0.968·25-s + 0.923·26-s + 0.308·27-s − 0.0512·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1502 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1502 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.322104932\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.322104932\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 751 | \( 1 + T \) |
good | 3 | \( 1 - 2.57T + 3T^{2} \) |
| 5 | \( 1 + 0.396T + 5T^{2} \) |
| 7 | \( 1 + 0.271T + 7T^{2} \) |
| 11 | \( 1 + 3.20T + 11T^{2} \) |
| 13 | \( 1 - 4.71T + 13T^{2} \) |
| 17 | \( 1 - 6.19T + 17T^{2} \) |
| 19 | \( 1 - 1.43T + 19T^{2} \) |
| 23 | \( 1 + 1.57T + 23T^{2} \) |
| 29 | \( 1 - 5.70T + 29T^{2} \) |
| 31 | \( 1 - 3.42T + 31T^{2} \) |
| 37 | \( 1 + 9.40T + 37T^{2} \) |
| 41 | \( 1 + 0.165T + 41T^{2} \) |
| 43 | \( 1 - 7.51T + 43T^{2} \) |
| 47 | \( 1 - 2.84T + 47T^{2} \) |
| 53 | \( 1 + 6.33T + 53T^{2} \) |
| 59 | \( 1 + 2.89T + 59T^{2} \) |
| 61 | \( 1 + 4.21T + 61T^{2} \) |
| 67 | \( 1 - 8.50T + 67T^{2} \) |
| 71 | \( 1 - 1.32T + 71T^{2} \) |
| 73 | \( 1 + 1.26T + 73T^{2} \) |
| 79 | \( 1 - 0.454T + 79T^{2} \) |
| 83 | \( 1 + 7.76T + 83T^{2} \) |
| 89 | \( 1 + 14.5T + 89T^{2} \) |
| 97 | \( 1 - 9.38T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.482934313399625763351903367199, −8.346595033580540002568001842467, −8.060687965391981573250696570368, −7.25835475974545773790994478142, −6.12738949235361523028752377644, −5.29421457150878055318472130792, −4.09679229632387760879100928336, −3.36354984580986178902496773710, −2.73694193582784129606419679019, −1.49228412830844895693333990469,
1.49228412830844895693333990469, 2.73694193582784129606419679019, 3.36354984580986178902496773710, 4.09679229632387760879100928336, 5.29421457150878055318472130792, 6.12738949235361523028752377644, 7.25835475974545773790994478142, 8.060687965391981573250696570368, 8.346595033580540002568001842467, 9.482934313399625763351903367199