| L(s) = 1 | + 2.47·5-s − 2.55·7-s − 0.669·11-s − 4.08·13-s − 6.44·17-s + 6.44·19-s − 2.82·23-s + 1.11·25-s + 4.35·29-s + 6.55·31-s − 6.32·35-s + 3.85·37-s − 0.788·41-s − 0.550·43-s + 2.82·47-s − 0.458·49-s + 3.64·53-s − 1.65·55-s + 5.65·59-s − 6.20·61-s − 10.1·65-s − 2.99·67-s − 5.11·71-s − 14.7·73-s + 1.71·77-s + 6.31·79-s + 0.907·83-s + ⋯ |
| L(s) = 1 | + 1.10·5-s − 0.966·7-s − 0.201·11-s − 1.13·13-s − 1.56·17-s + 1.47·19-s − 0.589·23-s + 0.223·25-s + 0.808·29-s + 1.17·31-s − 1.06·35-s + 0.634·37-s − 0.123·41-s − 0.0840·43-s + 0.412·47-s − 0.0654·49-s + 0.500·53-s − 0.223·55-s + 0.736·59-s − 0.794·61-s − 1.25·65-s − 0.366·67-s − 0.607·71-s − 1.72·73-s + 0.195·77-s + 0.711·79-s + 0.0996·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.784572071\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.784572071\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 2.47T + 5T^{2} \) |
| 7 | \( 1 + 2.55T + 7T^{2} \) |
| 11 | \( 1 + 0.669T + 11T^{2} \) |
| 13 | \( 1 + 4.08T + 13T^{2} \) |
| 17 | \( 1 + 6.44T + 17T^{2} \) |
| 19 | \( 1 - 6.44T + 19T^{2} \) |
| 23 | \( 1 + 2.82T + 23T^{2} \) |
| 29 | \( 1 - 4.35T + 29T^{2} \) |
| 31 | \( 1 - 6.55T + 31T^{2} \) |
| 37 | \( 1 - 3.85T + 37T^{2} \) |
| 41 | \( 1 + 0.788T + 41T^{2} \) |
| 43 | \( 1 + 0.550T + 43T^{2} \) |
| 47 | \( 1 - 2.82T + 47T^{2} \) |
| 53 | \( 1 - 3.64T + 53T^{2} \) |
| 59 | \( 1 - 5.65T + 59T^{2} \) |
| 61 | \( 1 + 6.20T + 61T^{2} \) |
| 67 | \( 1 + 2.99T + 67T^{2} \) |
| 71 | \( 1 + 5.11T + 71T^{2} \) |
| 73 | \( 1 + 14.7T + 73T^{2} \) |
| 79 | \( 1 - 6.31T + 79T^{2} \) |
| 83 | \( 1 - 0.907T + 83T^{2} \) |
| 89 | \( 1 - 6.31T + 89T^{2} \) |
| 97 | \( 1 - 12.6T + 97T^{2} \) |
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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
−7.55945529800925812070497742054, −6.97839774333251392411305471057, −6.25873323110985016690645674099, −5.84648496911009359436164297013, −4.93482217261117851180757813543, −4.39338404251876167510254103809, −3.20576639931142461379748704250, −2.60555421751559203659579691778, −1.91696745507428839281951573818, −0.61663973554084814704393437465,
0.61663973554084814704393437465, 1.91696745507428839281951573818, 2.60555421751559203659579691778, 3.20576639931142461379748704250, 4.39338404251876167510254103809, 4.93482217261117851180757813543, 5.84648496911009359436164297013, 6.25873323110985016690645674099, 6.97839774333251392411305471057, 7.55945529800925812070497742054
