| L(s) = 1 | + 2.42·2-s − 2.50·3-s + 3.90·4-s − 6.07·6-s + 7-s + 4.62·8-s + 3.25·9-s + 11-s − 9.76·12-s + 2.18·13-s + 2.42·14-s + 3.43·16-s + 1.02·17-s + 7.90·18-s − 5.00·19-s − 2.50·21-s + 2.42·22-s + 7.86·23-s − 11.5·24-s + 5.31·26-s − 0.628·27-s + 3.90·28-s − 0.713·29-s + 6.58·31-s − 0.902·32-s − 2.50·33-s + 2.48·34-s + ⋯ |
| L(s) = 1 | + 1.71·2-s − 1.44·3-s + 1.95·4-s − 2.48·6-s + 0.377·7-s + 1.63·8-s + 1.08·9-s + 0.301·11-s − 2.81·12-s + 0.607·13-s + 0.649·14-s + 0.859·16-s + 0.247·17-s + 1.86·18-s − 1.14·19-s − 0.545·21-s + 0.518·22-s + 1.63·23-s − 2.36·24-s + 1.04·26-s − 0.120·27-s + 0.737·28-s − 0.132·29-s + 1.18·31-s − 0.159·32-s − 0.435·33-s + 0.426·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.332858703\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.332858703\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 - 2.42T + 2T^{2} \) |
| 3 | \( 1 + 2.50T + 3T^{2} \) |
| 13 | \( 1 - 2.18T + 13T^{2} \) |
| 17 | \( 1 - 1.02T + 17T^{2} \) |
| 19 | \( 1 + 5.00T + 19T^{2} \) |
| 23 | \( 1 - 7.86T + 23T^{2} \) |
| 29 | \( 1 + 0.713T + 29T^{2} \) |
| 31 | \( 1 - 6.58T + 31T^{2} \) |
| 37 | \( 1 - 10.3T + 37T^{2} \) |
| 41 | \( 1 + 3.45T + 41T^{2} \) |
| 43 | \( 1 - 10.8T + 43T^{2} \) |
| 47 | \( 1 + 8.10T + 47T^{2} \) |
| 53 | \( 1 + 4.48T + 53T^{2} \) |
| 59 | \( 1 - 4.20T + 59T^{2} \) |
| 61 | \( 1 + 6.44T + 61T^{2} \) |
| 67 | \( 1 - 10.6T + 67T^{2} \) |
| 71 | \( 1 + 5.41T + 71T^{2} \) |
| 73 | \( 1 - 7.84T + 73T^{2} \) |
| 79 | \( 1 - 4.86T + 79T^{2} \) |
| 83 | \( 1 - 8.48T + 83T^{2} \) |
| 89 | \( 1 - 13.4T + 89T^{2} \) |
| 97 | \( 1 + 10.5T + 97T^{2} \) |
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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
−9.326030497766847622304567596889, −8.175892740014655326039587273811, −7.07560088052244719232504844959, −6.32553718099677946018792963855, −6.00853988035447990329707988469, −4.98675310936446044291672038511, −4.62611780587208592445249011438, −3.67822008066259434235765623189, −2.51753757387752502326300289242, −1.08821370788574103828201484139,
1.08821370788574103828201484139, 2.51753757387752502326300289242, 3.67822008066259434235765623189, 4.62611780587208592445249011438, 4.98675310936446044291672038511, 6.00853988035447990329707988469, 6.32553718099677946018792963855, 7.07560088052244719232504844959, 8.175892740014655326039587273811, 9.326030497766847622304567596889
