| L(s) = 1 | + 3-s + 0.502i·5-s + 3.10i·7-s + 9-s + i·11-s + (−3.60 − 0.0393i)13-s + 0.502i·15-s + 5.57·17-s − 1.43i·19-s + 3.10i·21-s + 0.536·23-s + 4.74·25-s + 27-s − 4.85·29-s + 6.70i·31-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.224i·5-s + 1.17i·7-s + 0.333·9-s + 0.301i·11-s + (−0.999 − 0.0109i)13-s + 0.129i·15-s + 1.35·17-s − 0.329i·19-s + 0.677i·21-s + 0.111·23-s + 0.949·25-s + 0.192·27-s − 0.901·29-s + 1.20i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1716 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0109 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1716 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0109 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.864533451\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.864533451\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 11 | \( 1 - iT \) |
| 13 | \( 1 + (3.60 + 0.0393i)T \) |
| good | 5 | \( 1 - 0.502iT - 5T^{2} \) |
| 7 | \( 1 - 3.10iT - 7T^{2} \) |
| 17 | \( 1 - 5.57T + 17T^{2} \) |
| 19 | \( 1 + 1.43iT - 19T^{2} \) |
| 23 | \( 1 - 0.536T + 23T^{2} \) |
| 29 | \( 1 + 4.85T + 29T^{2} \) |
| 31 | \( 1 - 6.70iT - 31T^{2} \) |
| 37 | \( 1 - 3.89iT - 37T^{2} \) |
| 41 | \( 1 - 8.72iT - 41T^{2} \) |
| 43 | \( 1 + 10.7T + 43T^{2} \) |
| 47 | \( 1 - 12.3iT - 47T^{2} \) |
| 53 | \( 1 + 3.25T + 53T^{2} \) |
| 59 | \( 1 + 7.87iT - 59T^{2} \) |
| 61 | \( 1 - 6.52T + 61T^{2} \) |
| 67 | \( 1 + 9.66iT - 67T^{2} \) |
| 71 | \( 1 - 11.1iT - 71T^{2} \) |
| 73 | \( 1 + 3.10iT - 73T^{2} \) |
| 79 | \( 1 + 12.7T + 79T^{2} \) |
| 83 | \( 1 - 9.17iT - 83T^{2} \) |
| 89 | \( 1 - 5.76iT - 89T^{2} \) |
| 97 | \( 1 - 3.69iT - 97T^{2} \) |
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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
−9.605334827697748216109963174820, −8.713809733342871823706250409124, −8.028238885088950131855690211704, −7.22327485969237589246247297759, −6.39883537711294967562640633426, −5.31579746892489771376865325626, −4.73584132402638475188939887866, −3.27059296429437513731086783687, −2.73372325470827531848660744357, −1.56626276687590123880256751705,
0.65993151766396479440964575284, 1.98248346230115577873967770110, 3.27047340398135411002805316987, 3.95028650381403947074397030874, 4.95204618347953313893739028903, 5.82039080072202987438247083589, 7.17751550624925629425550059752, 7.38064425244623335254322493198, 8.319003793558444289079572013582, 9.108304427659445209722026839527
