L(s) = 1 | + (0.5 − 0.363i)2-s + (0.5 + 1.53i)3-s + (−0.190 + 0.587i)4-s + (−0.809 − 0.587i)5-s + (0.809 + 0.587i)6-s + (0.309 + 0.951i)8-s + (−1.30 + 0.951i)9-s − 0.618·10-s + (−0.809 + 0.587i)11-s − 12-s + (0.5 − 0.363i)13-s + (0.5 − 1.53i)15-s + (−0.309 + 0.951i)18-s + (0.309 + 0.951i)19-s + (0.5 − 0.363i)20-s + ⋯ |
L(s) = 1 | + (0.5 − 0.363i)2-s + (0.5 + 1.53i)3-s + (−0.190 + 0.587i)4-s + (−0.809 − 0.587i)5-s + (0.809 + 0.587i)6-s + (0.309 + 0.951i)8-s + (−1.30 + 0.951i)9-s − 0.618·10-s + (−0.809 + 0.587i)11-s − 12-s + (0.5 − 0.363i)13-s + (0.5 − 1.53i)15-s + (−0.309 + 0.951i)18-s + (0.309 + 0.951i)19-s + (0.5 − 0.363i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.286 - 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.286 - 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.198571997\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.198571997\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (0.809 + 0.587i)T \) |
| 11 | \( 1 + (0.809 - 0.587i)T \) |
| 19 | \( 1 + (-0.309 - 0.951i)T \) |
good | 2 | \( 1 + (-0.5 + 0.363i)T + (0.309 - 0.951i)T^{2} \) |
| 3 | \( 1 + (-0.5 - 1.53i)T + (-0.809 + 0.587i)T^{2} \) |
| 7 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 13 | \( 1 + (-0.5 + 0.363i)T + (0.309 - 0.951i)T^{2} \) |
| 17 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 31 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 37 | \( 1 + (-0.5 + 1.53i)T + (-0.809 - 0.587i)T^{2} \) |
| 41 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 53 | \( 1 + (-1.61 + 1.17i)T + (0.309 - 0.951i)T^{2} \) |
| 59 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 61 | \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \) |
| 67 | \( 1 - 1.61T + T^{2} \) |
| 71 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 73 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 79 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 83 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (-0.5 + 0.363i)T + (0.309 - 0.951i)T^{2} \) |
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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
−10.37096003960788698513573144638, −9.584684134114830841235381236687, −8.646014186531621616246005611078, −8.180380394816773625685766847038, −7.37971833560627100841222513019, −5.45917042125901791129522921265, −4.93453785578919682234408112357, −3.90919550956115731307420495052, −3.66140989280764588052589120736, −2.46250025927112361202436008661,
0.950504374774299536713271929731, 2.47685343319124064022258699028, 3.46548776312403070719615115986, 4.71536712394089586724552567149, 5.87189191028273826929244737667, 6.65195601822996493768912045540, 7.17956378520416800575798428200, 8.044013992557834281487262072729, 8.686776005902690497707155373100, 9.847301922104327091171063081802