L(s) = 1 | + (0.658 − 0.600i)2-s + (−0.0189 + 0.204i)4-s + (0.273 + 0.961i)7-s + (0.647 + 0.857i)8-s + (−0.445 − 0.895i)9-s + (−0.136 + 1.47i)11-s + (0.757 + 0.469i)14-s + (0.739 + 0.138i)16-s + (−0.831 − 0.322i)18-s + (0.794 + 1.05i)22-s + (0.0675 − 0.361i)23-s + (−0.0922 − 0.995i)25-s + (−0.201 + 0.0377i)28-s + (0.247 − 1.32i)29-s + (−0.343 + 0.212i)32-s + ⋯ |
L(s) = 1 | + (0.658 − 0.600i)2-s + (−0.0189 + 0.204i)4-s + (0.273 + 0.961i)7-s + (0.647 + 0.857i)8-s + (−0.445 − 0.895i)9-s + (−0.136 + 1.47i)11-s + (0.757 + 0.469i)14-s + (0.739 + 0.138i)16-s + (−0.831 − 0.322i)18-s + (0.794 + 1.05i)22-s + (0.0675 − 0.361i)23-s + (−0.0922 − 0.995i)25-s + (−0.201 + 0.0377i)28-s + (0.247 − 1.32i)29-s + (−0.343 + 0.212i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 959 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 - 0.181i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 959 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 - 0.181i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.407981284\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.407981284\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (-0.273 - 0.961i)T \) |
| 137 | \( 1 + T \) |
good | 2 | \( 1 + (-0.658 + 0.600i)T + (0.0922 - 0.995i)T^{2} \) |
| 3 | \( 1 + (0.445 + 0.895i)T^{2} \) |
| 5 | \( 1 + (0.0922 + 0.995i)T^{2} \) |
| 11 | \( 1 + (0.136 - 1.47i)T + (-0.982 - 0.183i)T^{2} \) |
| 13 | \( 1 + (-0.850 + 0.526i)T^{2} \) |
| 17 | \( 1 + (0.273 + 0.961i)T^{2} \) |
| 19 | \( 1 + (-0.739 - 0.673i)T^{2} \) |
| 23 | \( 1 + (-0.0675 + 0.361i)T + (-0.932 - 0.361i)T^{2} \) |
| 29 | \( 1 + (-0.247 + 1.32i)T + (-0.932 - 0.361i)T^{2} \) |
| 31 | \( 1 + (0.739 - 0.673i)T^{2} \) |
| 37 | \( 1 - 0.184T + T^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + (-0.132 + 0.342i)T + (-0.739 - 0.673i)T^{2} \) |
| 47 | \( 1 + (-0.602 + 0.798i)T^{2} \) |
| 53 | \( 1 + (-0.380 + 0.981i)T + (-0.739 - 0.673i)T^{2} \) |
| 59 | \( 1 + (0.602 - 0.798i)T^{2} \) |
| 61 | \( 1 + (0.602 + 0.798i)T^{2} \) |
| 67 | \( 1 + (-1.72 + 0.489i)T + (0.850 - 0.526i)T^{2} \) |
| 71 | \( 1 + (1.98 - 0.183i)T + (0.982 - 0.183i)T^{2} \) |
| 73 | \( 1 + (0.850 + 0.526i)T^{2} \) |
| 79 | \( 1 + (0.380 + 0.614i)T + (-0.445 + 0.895i)T^{2} \) |
| 83 | \( 1 + (-0.273 + 0.961i)T^{2} \) |
| 89 | \( 1 + (0.0922 + 0.995i)T^{2} \) |
| 97 | \( 1 + (-0.982 - 0.183i)T^{2} \) |
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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
−10.30569977547421852584311520509, −9.498800290759996152481825028202, −8.580092695307411352860337572945, −7.87660688393214193450615078888, −6.76608437867381722432931769343, −5.75968138826759934566512541877, −4.77873454800551382524972450009, −4.02641826642534486257199201786, −2.79316767997292805511927804283, −2.05488769182451142071173634315,
1.29554734125878117701305106424, 3.11913183834086991641241350184, 4.13229213431788235095653279404, 5.17541118482741746101220377141, 5.69693259551834684783126393301, 6.75311216145146555810584167819, 7.54429382799342975934011112541, 8.322127880325172089722770989964, 9.342372421411527344854975672072, 10.53357527127165995217390406319