| L(s) = 1 | + 2-s + 4-s − 3.71i·5-s − 7-s + 8-s − 3.71i·10-s − 1.06i·11-s − 4.70i·13-s − 14-s + 16-s − 1.06i·17-s + (−3.96 − 1.82i)19-s − 3.71i·20-s − 1.06i·22-s + 5.31i·23-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 1.66i·5-s − 0.377·7-s + 0.353·8-s − 1.17i·10-s − 0.322i·11-s − 1.30i·13-s − 0.267·14-s + 0.250·16-s − 0.259i·17-s + (−0.908 − 0.417i)19-s − 0.831i·20-s − 0.227i·22-s + 1.10i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.865 + 0.500i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.865 + 0.500i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.886483477\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.886483477\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 + (3.96 + 1.82i)T \) |
| good | 5 | \( 1 + 3.71iT - 5T^{2} \) |
| 11 | \( 1 + 1.06iT - 11T^{2} \) |
| 13 | \( 1 + 4.70iT - 13T^{2} \) |
| 17 | \( 1 + 1.06iT - 17T^{2} \) |
| 23 | \( 1 - 5.31iT - 23T^{2} \) |
| 29 | \( 1 + 7.83T + 29T^{2} \) |
| 31 | \( 1 - 2.65iT - 31T^{2} \) |
| 37 | \( 1 + 3.19iT - 37T^{2} \) |
| 41 | \( 1 - 11.5T + 41T^{2} \) |
| 43 | \( 1 + 10.6T + 43T^{2} \) |
| 47 | \( 1 - 9.41iT - 47T^{2} \) |
| 53 | \( 1 - 8.68T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 11.0T + 61T^{2} \) |
| 67 | \( 1 + 4.10iT - 67T^{2} \) |
| 71 | \( 1 + 5.14T + 71T^{2} \) |
| 73 | \( 1 + 7.00T + 73T^{2} \) |
| 79 | \( 1 + 3.66iT - 79T^{2} \) |
| 83 | \( 1 + 9.63iT - 83T^{2} \) |
| 89 | \( 1 + 6.44T + 89T^{2} \) |
| 97 | \( 1 + 13.1iT - 97T^{2} \) |
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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
−8.649876376906800451727810239869, −7.893283345602782297045664373658, −7.14382232455596885719164129861, −5.82692283378151177644682875020, −5.58303037434761786619206034892, −4.68655711738775185047305057222, −3.90922587123462360444355007027, −2.95186727964110910430714614947, −1.65903250203909076752698646411, −0.46246273237944113459778946132,
2.01925544314924241366141534185, 2.59380860654455351872952624175, 3.77203904949534447832322275022, 4.16063137714487533318028503440, 5.48819169161154559881318916503, 6.42378991715546057757498983209, 6.72547720855235248861354834741, 7.40821406723655441630569408693, 8.409012060472426514076062656710, 9.454576681121576148385811580177
