| L(s) = 1 | − 2.41·2-s + 3.82·4-s − 5-s − 7-s − 4.41·8-s + 2.41·10-s − 0.414·11-s + 2.41·13-s + 2.41·14-s + 2.99·16-s − 4.82·17-s + 1.58·19-s − 3.82·20-s + 0.999·22-s + 0.828·23-s + 25-s − 5.82·26-s − 3.82·28-s + 4·29-s + 6·31-s + 1.58·32-s + 11.6·34-s + 35-s − 8.48·37-s − 3.82·38-s + 4.41·40-s + 7.82·41-s + ⋯ |
| L(s) = 1 | − 1.70·2-s + 1.91·4-s − 0.447·5-s − 0.377·7-s − 1.56·8-s + 0.763·10-s − 0.124·11-s + 0.669·13-s + 0.645·14-s + 0.749·16-s − 1.17·17-s + 0.363·19-s − 0.856·20-s + 0.213·22-s + 0.172·23-s + 0.200·25-s − 1.14·26-s − 0.723·28-s + 0.742·29-s + 1.07·31-s + 0.280·32-s + 1.99·34-s + 0.169·35-s − 1.39·37-s − 0.621·38-s + 0.697·40-s + 1.22·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5389881282\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5389881282\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| good | 2 | \( 1 + 2.41T + 2T^{2} \) |
| 11 | \( 1 + 0.414T + 11T^{2} \) |
| 13 | \( 1 - 2.41T + 13T^{2} \) |
| 17 | \( 1 + 4.82T + 17T^{2} \) |
| 19 | \( 1 - 1.58T + 19T^{2} \) |
| 23 | \( 1 - 0.828T + 23T^{2} \) |
| 29 | \( 1 - 4T + 29T^{2} \) |
| 31 | \( 1 - 6T + 31T^{2} \) |
| 37 | \( 1 + 8.48T + 37T^{2} \) |
| 41 | \( 1 - 7.82T + 41T^{2} \) |
| 43 | \( 1 + 6.65T + 43T^{2} \) |
| 47 | \( 1 + 7.48T + 47T^{2} \) |
| 53 | \( 1 - 10.8T + 53T^{2} \) |
| 59 | \( 1 - 6.48T + 59T^{2} \) |
| 61 | \( 1 + 8.48T + 61T^{2} \) |
| 67 | \( 1 - 7.82T + 67T^{2} \) |
| 71 | \( 1 + 2T + 71T^{2} \) |
| 73 | \( 1 + 2.07T + 73T^{2} \) |
| 79 | \( 1 - 14.8T + 79T^{2} \) |
| 83 | \( 1 - 7.48T + 83T^{2} \) |
| 89 | \( 1 - 8.65T + 89T^{2} \) |
| 97 | \( 1 - 14.1T + 97T^{2} \) |
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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
−10.02792358613329028896778447612, −9.027239797356005786129952452974, −8.572370578576987369747785803198, −7.75991674089887446705496511930, −6.86695097279856306167237730752, −6.26244710271478449709489221025, −4.75650797871978601491027775130, −3.39046012594370553455550625182, −2.16753262769038523831216053910, −0.73557699812010142157301659130,
0.73557699812010142157301659130, 2.16753262769038523831216053910, 3.39046012594370553455550625182, 4.75650797871978601491027775130, 6.26244710271478449709489221025, 6.86695097279856306167237730752, 7.75991674089887446705496511930, 8.572370578576987369747785803198, 9.027239797356005786129952452974, 10.02792358613329028896778447612
