| L(s) = 1 | − 2.23·2-s + 3.00·4-s − 5-s − 2.23·8-s + 2.23·10-s + 2.47·11-s + 4.47·13-s − 0.999·16-s − 2·17-s − 6.47·19-s − 3.00·20-s − 5.52·22-s − 4·23-s + 25-s − 10.0·26-s + 2·29-s − 10.4·31-s + 6.70·32-s + 4.47·34-s + 10.9·37-s + 14.4·38-s + 2.23·40-s − 2·41-s − 8.94·43-s + 7.41·44-s + 8.94·46-s − 4.94·47-s + ⋯ |
| L(s) = 1 | − 1.58·2-s + 1.50·4-s − 0.447·5-s − 0.790·8-s + 0.707·10-s + 0.745·11-s + 1.24·13-s − 0.249·16-s − 0.485·17-s − 1.48·19-s − 0.670·20-s − 1.17·22-s − 0.834·23-s + 0.200·25-s − 1.96·26-s + 0.371·29-s − 1.88·31-s + 1.18·32-s + 0.766·34-s + 1.79·37-s + 2.34·38-s + 0.353·40-s − 0.312·41-s − 1.36·43-s + 1.11·44-s + 1.31·46-s − 0.721·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 2.23T + 2T^{2} \) |
| 11 | \( 1 - 2.47T + 11T^{2} \) |
| 13 | \( 1 - 4.47T + 13T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 + 6.47T + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + 10.4T + 31T^{2} \) |
| 37 | \( 1 - 10.9T + 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + 8.94T + 43T^{2} \) |
| 47 | \( 1 + 4.94T + 47T^{2} \) |
| 53 | \( 1 - 12.4T + 53T^{2} \) |
| 59 | \( 1 - 8.94T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 + 14.4T + 71T^{2} \) |
| 73 | \( 1 - 3.52T + 73T^{2} \) |
| 79 | \( 1 + 4.94T + 79T^{2} \) |
| 83 | \( 1 - 0.944T + 83T^{2} \) |
| 89 | \( 1 + 2T + 89T^{2} \) |
| 97 | \( 1 - 0.472T + 97T^{2} \) |
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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
−8.527262459754565214760602549472, −8.321825935656811848825267409219, −7.26748860243838577988284499505, −6.60844702447583279411046975298, −5.90288449891525962799111143183, −4.40478139629409449802217709687, −3.70179222009514294502738572296, −2.25995413443724394857843608840, −1.32142969098230282350593928673, 0,
1.32142969098230282350593928673, 2.25995413443724394857843608840, 3.70179222009514294502738572296, 4.40478139629409449802217709687, 5.90288449891525962799111143183, 6.60844702447583279411046975298, 7.26748860243838577988284499505, 8.321825935656811848825267409219, 8.527262459754565214760602549472
