L(s) = 1 | + 2-s + (−1.36 + 1.06i)3-s + 4-s + (2.16 − 0.546i)5-s + (−1.36 + 1.06i)6-s − 1.97i·7-s + 8-s + (0.733 − 2.90i)9-s + (2.16 − 0.546i)10-s − 0.980·11-s + (−1.36 + 1.06i)12-s + 4.39·13-s − 1.97i·14-s + (−2.38 + 3.05i)15-s + 16-s − 2.02i·17-s + ⋯ |
L(s) = 1 | + 0.707·2-s + (−0.788 + 0.614i)3-s + 0.5·4-s + (0.969 − 0.244i)5-s + (−0.557 + 0.434i)6-s − 0.746i·7-s + 0.353·8-s + (0.244 − 0.969i)9-s + (0.685 − 0.172i)10-s − 0.295·11-s + (−0.394 + 0.307i)12-s + 1.21·13-s − 0.527i·14-s + (−0.614 + 0.788i)15-s + 0.250·16-s − 0.490i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.951 + 0.308i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.951 + 0.308i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.24419 - 0.354668i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.24419 - 0.354668i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 + (1.36 - 1.06i)T \) |
| 5 | \( 1 + (-2.16 + 0.546i)T \) |
| 31 | \( 1 + (1.90 + 5.23i)T \) |
good | 7 | \( 1 + 1.97iT - 7T^{2} \) |
| 11 | \( 1 + 0.980T + 11T^{2} \) |
| 13 | \( 1 - 4.39T + 13T^{2} \) |
| 17 | \( 1 + 2.02iT - 17T^{2} \) |
| 19 | \( 1 + 3.72T + 19T^{2} \) |
| 23 | \( 1 + 6.09iT - 23T^{2} \) |
| 29 | \( 1 + 2.54T + 29T^{2} \) |
| 37 | \( 1 - 9.44T + 37T^{2} \) |
| 41 | \( 1 - 12.5iT - 41T^{2} \) |
| 43 | \( 1 + 3.33T + 43T^{2} \) |
| 47 | \( 1 - 11.6T + 47T^{2} \) |
| 53 | \( 1 - 7.67iT - 53T^{2} \) |
| 59 | \( 1 - 0.348iT - 59T^{2} \) |
| 61 | \( 1 - 5.15iT - 61T^{2} \) |
| 67 | \( 1 + 1.03iT - 67T^{2} \) |
| 71 | \( 1 - 8.70iT - 71T^{2} \) |
| 73 | \( 1 + 5.88T + 73T^{2} \) |
| 79 | \( 1 + 8.91iT - 79T^{2} \) |
| 83 | \( 1 - 8.99iT - 83T^{2} \) |
| 89 | \( 1 + 10.3T + 89T^{2} \) |
| 97 | \( 1 + 17.1iT - 97T^{2} \) |
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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
−10.26102273054348496913117733983, −9.428175709246105910282639709170, −8.449414006093208873887861990780, −7.13990089159699963712191602684, −6.16772690162147122953891353512, −5.81543531125738370749393854654, −4.60325695983398642883153072899, −4.08880918077521838740635858580, −2.67537039449436635959070455531, −1.04923927017039950920913203208,
1.54832930777560291504329868651, 2.44330331918430482411666443280, 3.84221242970476450669101399766, 5.26175373295357812038411697009, 5.75513317861557080210851460684, 6.37087256631438218055224497339, 7.25200023560862409738546115014, 8.364395569213707974025324168988, 9.297174267692651531676103477617, 10.54720819869043034065889921435