L(s) = 1 | + (0.587 − 0.809i)4-s + (−0.809 − 0.587i)5-s + (0.309 + 1.95i)7-s + (0.951 + 0.309i)9-s + (0.587 + 0.809i)11-s + (−0.309 − 0.951i)16-s + (−0.412 + 0.809i)17-s + (0.809 − 0.587i)19-s + (−0.951 + 0.309i)20-s + (−1.26 − 1.26i)23-s + (0.309 + 0.951i)25-s + (1.76 + 0.896i)28-s + (0.896 − 1.76i)35-s + (0.809 − 0.587i)36-s + (1.26 − 1.26i)43-s + 44-s + ⋯ |
L(s) = 1 | + (0.587 − 0.809i)4-s + (−0.809 − 0.587i)5-s + (0.309 + 1.95i)7-s + (0.951 + 0.309i)9-s + (0.587 + 0.809i)11-s + (−0.309 − 0.951i)16-s + (−0.412 + 0.809i)17-s + (0.809 − 0.587i)19-s + (−0.951 + 0.309i)20-s + (−1.26 − 1.26i)23-s + (0.309 + 0.951i)25-s + (1.76 + 0.896i)28-s + (0.896 − 1.76i)35-s + (0.809 − 0.587i)36-s + (1.26 − 1.26i)43-s + 44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0527i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0527i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.141546430\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.141546430\) |
\(L(1)\) |
|
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 | \( 1 + (-0.587 - 0.809i)T \) |
| 19 | \( 1 + (-0.809 + 0.587i)T \) |
good | 2 | \( 1 + (-0.587 + 0.809i)T^{2} \) |
| 3 | \( 1 + (-0.951 - 0.309i)T^{2} \) |
| 7 | \( 1 + (-0.309 - 1.95i)T + (-0.951 + 0.309i)T^{2} \) |
| 13 | \( 1 + (0.587 - 0.809i)T^{2} \) |
| 17 | \( 1 + (0.412 - 0.809i)T + (-0.587 - 0.809i)T^{2} \) |
| 23 | \( 1 + (1.26 + 1.26i)T + iT^{2} \) |
| 29 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 31 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 37 | \( 1 + (-0.951 + 0.309i)T^{2} \) |
| 41 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 43 | \( 1 + (-1.26 + 1.26i)T - iT^{2} \) |
| 47 | \( 1 + (-0.142 + 0.896i)T + (-0.951 - 0.309i)T^{2} \) |
| 53 | \( 1 + (-0.587 + 0.809i)T^{2} \) |
| 59 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 61 | \( 1 + (0.587 - 0.190i)T + (0.809 - 0.587i)T^{2} \) |
| 67 | \( 1 + iT^{2} \) |
| 71 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 73 | \( 1 + (1.39 - 0.221i)T + (0.951 - 0.309i)T^{2} \) |
| 79 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 83 | \( 1 + (-0.278 - 0.142i)T + (0.587 + 0.809i)T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + (0.587 - 0.809i)T^{2} \) |
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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
−10.08098680512474593830366414118, −9.233935142769473312081585900198, −8.608141928062725685505363404747, −7.62993732885130493723155809680, −6.72375284973211732524451478424, −5.79750718180171649061335540023, −4.97431489913492046191513615541, −4.16719951670227916784774001988, −2.45028556393840762608289219288, −1.64033749352324593505007525334,
1.30790359288508258780122014736, 3.16636252403556442422088020047, 3.87805617599714412630245613422, 4.36989187883394538345081472708, 6.23292439865907912634744807356, 7.03579202627393213648223197400, 7.60320820713597594464349574950, 7.954664044607568525834951024361, 9.398882470347733718969361366619, 10.28449498101639950449914673728