| L(s) = 1 | − 2.82·5-s − 3.46·7-s − 11-s − 6.89·13-s + 6.29·17-s + 6.29·19-s + 4.89·23-s + 3.00·25-s + 0.635·29-s + 5.65·31-s + 9.79·35-s + 7.79·37-s − 0.635·41-s − 0.635·43-s − 8.89·47-s + 4.99·49-s + 9.75·53-s + 2.82·55-s − 13.7·59-s − 1.10·61-s + 19.5·65-s + 0.898·71-s + 6·73-s + 3.46·77-s + 16.0·79-s − 13.7·83-s − 17.7·85-s + ⋯ |
| L(s) = 1 | − 1.26·5-s − 1.30·7-s − 0.301·11-s − 1.91·13-s + 1.52·17-s + 1.44·19-s + 1.02·23-s + 0.600·25-s + 0.118·29-s + 1.01·31-s + 1.65·35-s + 1.28·37-s − 0.0992·41-s − 0.0969·43-s − 1.29·47-s + 0.714·49-s + 1.34·53-s + 0.381·55-s − 1.79·59-s − 0.140·61-s + 2.42·65-s + 0.106·71-s + 0.702·73-s + 0.394·77-s + 1.80·79-s − 1.51·83-s − 1.93·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 + 2.82T + 5T^{2} \) |
| 7 | \( 1 + 3.46T + 7T^{2} \) |
| 13 | \( 1 + 6.89T + 13T^{2} \) |
| 17 | \( 1 - 6.29T + 17T^{2} \) |
| 19 | \( 1 - 6.29T + 19T^{2} \) |
| 23 | \( 1 - 4.89T + 23T^{2} \) |
| 29 | \( 1 - 0.635T + 29T^{2} \) |
| 31 | \( 1 - 5.65T + 31T^{2} \) |
| 37 | \( 1 - 7.79T + 37T^{2} \) |
| 41 | \( 1 + 0.635T + 41T^{2} \) |
| 43 | \( 1 + 0.635T + 43T^{2} \) |
| 47 | \( 1 + 8.89T + 47T^{2} \) |
| 53 | \( 1 - 9.75T + 53T^{2} \) |
| 59 | \( 1 + 13.7T + 59T^{2} \) |
| 61 | \( 1 + 1.10T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 - 0.898T + 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 - 16.0T + 79T^{2} \) |
| 83 | \( 1 + 13.7T + 83T^{2} \) |
| 89 | \( 1 + 11.3T + 89T^{2} \) |
| 97 | \( 1 - 10T + 97T^{2} \) |
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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
−7.64927645648429773718353808804, −7.16662184462020300589963656537, −6.42086342731486467234505377396, −5.38474222210604246399857025307, −4.86187088890666413502310453848, −3.90637381489240985716811838704, −3.03651226187470840534569544656, −2.80757668647088730733281603563, −0.986716052472749583316732952479, 0,
0.986716052472749583316732952479, 2.80757668647088730733281603563, 3.03651226187470840534569544656, 3.90637381489240985716811838704, 4.86187088890666413502310453848, 5.38474222210604246399857025307, 6.42086342731486467234505377396, 7.16662184462020300589963656537, 7.64927645648429773718353808804
