L(s) = 1 | − 1.51i·3-s − 1.74i·5-s − 0.991i·7-s + 0.701·9-s + 3.70i·11-s + 3.69·13-s − 2.65·15-s + (1.53 + 3.82i)17-s + 0.316·19-s − 1.50·21-s − 5.72i·23-s + 1.93·25-s − 5.61i·27-s + 7.42i·29-s − 6.07i·31-s + ⋯ |
L(s) = 1 | − 0.875i·3-s − 0.782i·5-s − 0.374i·7-s + 0.233·9-s + 1.11i·11-s + 1.02·13-s − 0.684·15-s + (0.371 + 0.928i)17-s + 0.0726·19-s − 0.327·21-s − 1.19i·23-s + 0.387·25-s − 1.07i·27-s + 1.37i·29-s − 1.09i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.371 + 0.928i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.371 + 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.262437259\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.262437259\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 17 | \( 1 + (-1.53 - 3.82i)T \) |
| 59 | \( 1 + T \) |
good | 3 | \( 1 + 1.51iT - 3T^{2} \) |
| 5 | \( 1 + 1.74iT - 5T^{2} \) |
| 7 | \( 1 + 0.991iT - 7T^{2} \) |
| 11 | \( 1 - 3.70iT - 11T^{2} \) |
| 13 | \( 1 - 3.69T + 13T^{2} \) |
| 19 | \( 1 - 0.316T + 19T^{2} \) |
| 23 | \( 1 + 5.72iT - 23T^{2} \) |
| 29 | \( 1 - 7.42iT - 29T^{2} \) |
| 31 | \( 1 + 6.07iT - 31T^{2} \) |
| 37 | \( 1 - 5.20iT - 37T^{2} \) |
| 41 | \( 1 - 1.04iT - 41T^{2} \) |
| 43 | \( 1 - 0.358T + 43T^{2} \) |
| 47 | \( 1 - 5.00T + 47T^{2} \) |
| 53 | \( 1 - 8.46T + 53T^{2} \) |
| 61 | \( 1 - 7.21iT - 61T^{2} \) |
| 67 | \( 1 - 8.81T + 67T^{2} \) |
| 71 | \( 1 - 9.43iT - 71T^{2} \) |
| 73 | \( 1 - 2.28iT - 73T^{2} \) |
| 79 | \( 1 - 10.9iT - 79T^{2} \) |
| 83 | \( 1 + 0.243T + 83T^{2} \) |
| 89 | \( 1 + 0.105T + 89T^{2} \) |
| 97 | \( 1 + 11.1iT - 97T^{2} \) |
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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
−8.424637504902560004439688337753, −7.46854627313365054622761637112, −6.94527759884988692949283855777, −6.23751205985305428663386372730, −5.37035124339943120114635380668, −4.40614776105668789871804162906, −3.91457845308374365261717357295, −2.53268944129659030157652319640, −1.52079050571456630571794188430, −0.913754297561131106199907409477,
0.951899200078816520328895022533, 2.38863600742872005066773071099, 3.43440189354378354440100247109, 3.69235110717964273124844096119, 4.87436659542155891763325939965, 5.61544786767436828937584888981, 6.25373914175139850075888747129, 7.15239821144479426789464945712, 7.82412521252318086643476622935, 8.856021507963737460949245139281