| L(s) = 1 | + 2.41·2-s + 3.82·4-s + 2.82·5-s + 4.41·8-s + 6.82·10-s + 2·11-s + 13-s + 2.99·16-s − 3.65·17-s − 2.82·19-s + 10.8·20-s + 4.82·22-s + 4·23-s + 3.00·25-s + 2.41·26-s − 2·29-s + 6.82·31-s − 1.58·32-s − 8.82·34-s + 3.65·37-s − 6.82·38-s + 12.4·40-s + 10.8·41-s + 9.65·43-s + 7.65·44-s + 9.65·46-s − 0.343·47-s + ⋯ |
| L(s) = 1 | + 1.70·2-s + 1.91·4-s + 1.26·5-s + 1.56·8-s + 2.15·10-s + 0.603·11-s + 0.277·13-s + 0.749·16-s − 0.886·17-s − 0.648·19-s + 2.42·20-s + 1.02·22-s + 0.834·23-s + 0.600·25-s + 0.473·26-s − 0.371·29-s + 1.22·31-s − 0.280·32-s − 1.51·34-s + 0.601·37-s − 1.10·38-s + 1.97·40-s + 1.69·41-s + 1.47·43-s + 1.15·44-s + 1.42·46-s − 0.0500·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5733 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5733 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.740026712\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.740026712\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 - 2.41T + 2T^{2} \) |
| 5 | \( 1 - 2.82T + 5T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 17 | \( 1 + 3.65T + 17T^{2} \) |
| 19 | \( 1 + 2.82T + 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 - 6.82T + 31T^{2} \) |
| 37 | \( 1 - 3.65T + 37T^{2} \) |
| 41 | \( 1 - 10.8T + 41T^{2} \) |
| 43 | \( 1 - 9.65T + 43T^{2} \) |
| 47 | \( 1 + 0.343T + 47T^{2} \) |
| 53 | \( 1 - 2T + 53T^{2} \) |
| 59 | \( 1 + 3.65T + 59T^{2} \) |
| 61 | \( 1 - 9.31T + 61T^{2} \) |
| 67 | \( 1 - 1.17T + 67T^{2} \) |
| 71 | \( 1 + 2T + 71T^{2} \) |
| 73 | \( 1 + 11.6T + 73T^{2} \) |
| 79 | \( 1 - 11.3T + 79T^{2} \) |
| 83 | \( 1 + 7.65T + 83T^{2} \) |
| 89 | \( 1 - 9.17T + 89T^{2} \) |
| 97 | \( 1 - 7.65T + 97T^{2} \) |
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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
−7.912343558024960502711120491155, −6.90215418212367991319178299238, −6.41824659882695288477315050285, −5.91065514463364234418254550977, −5.27444703650678177051637943280, −4.40188137905306314854500878937, −3.95579579786575974973634972698, −2.73103931502552884079111428236, −2.33970686826880686347116639251, −1.25909092236126833295496297511,
1.25909092236126833295496297511, 2.33970686826880686347116639251, 2.73103931502552884079111428236, 3.95579579786575974973634972698, 4.40188137905306314854500878937, 5.27444703650678177051637943280, 5.91065514463364234418254550977, 6.41824659882695288477315050285, 6.90215418212367991319178299238, 7.912343558024960502711120491155
