| L(s) = 1 | − i·3-s + (−1.75 − 1.38i)5-s − 9-s − 1.50·11-s + i·13-s + (−1.38 + 1.75i)15-s + 2.72i·17-s + 0.726·19-s + 4.72i·23-s + (1.14 + 4.86i)25-s + i·27-s + 7.55·29-s + 3.00·31-s + 1.50i·33-s + 5.00i·37-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + (−0.783 − 0.621i)5-s − 0.333·9-s − 0.453·11-s + 0.277i·13-s + (−0.358 + 0.452i)15-s + 0.661i·17-s + 0.166·19-s + 0.985i·23-s + (0.228 + 0.973i)25-s + 0.192i·27-s + 1.40·29-s + 0.540·31-s + 0.261i·33-s + 0.823i·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.783 + 0.621i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.783 + 0.621i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.369375433\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.369375433\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 + (1.75 + 1.38i)T \) |
| 13 | \( 1 - iT \) |
| good | 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 + 1.50T + 11T^{2} \) |
| 17 | \( 1 - 2.72iT - 17T^{2} \) |
| 19 | \( 1 - 0.726T + 19T^{2} \) |
| 23 | \( 1 - 4.72iT - 23T^{2} \) |
| 29 | \( 1 - 7.55T + 29T^{2} \) |
| 31 | \( 1 - 3.00T + 31T^{2} \) |
| 37 | \( 1 - 5.00iT - 37T^{2} \) |
| 41 | \( 1 - 5.78T + 41T^{2} \) |
| 43 | \( 1 + 2.72iT - 43T^{2} \) |
| 47 | \( 1 + 10.2iT - 47T^{2} \) |
| 53 | \( 1 + 7.55iT - 53T^{2} \) |
| 59 | \( 1 + 12.5T + 59T^{2} \) |
| 61 | \( 1 - 6.28T + 61T^{2} \) |
| 67 | \( 1 + 12.5iT - 67T^{2} \) |
| 71 | \( 1 + 4.77T + 71T^{2} \) |
| 73 | \( 1 - 12.0iT - 73T^{2} \) |
| 79 | \( 1 - 5.27T + 79T^{2} \) |
| 83 | \( 1 + 7.78iT - 83T^{2} \) |
| 89 | \( 1 + 1.78T + 89T^{2} \) |
| 97 | \( 1 - 6iT - 97T^{2} \) |
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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
−8.421302608766991071759371647430, −7.955173554820651309057013525535, −7.21233845404225744217441788930, −6.44936986249266500869688333585, −5.52062340413919473029141197597, −4.77699109396742859052047280204, −3.88912970204085948270739178323, −2.99047286123964886112421061733, −1.79708542088910533601733190947, −0.71317089395745856639903613525,
0.69706277352882374359123859339, 2.61763409108892445073203585862, 3.02697237492869481377911624690, 4.22658777644963277224711219020, 4.66238542577320898374108318458, 5.74867537595910101144280752122, 6.51805259157415515274548803139, 7.40000020896567877945642469242, 7.977617394882667166333712971625, 8.742315897698293989163884126062
