| L(s) = 1 | − 5-s − 5.09·7-s − 1.25·11-s − 13-s + 7.09·17-s − 6.35·19-s + 5.61·23-s + 25-s + 4.35·29-s + 2·31-s + 5.09·35-s + 0.741·37-s − 0.741·41-s − 5.48·43-s + 5.48·47-s + 18.9·49-s − 6.74·53-s + 1.25·55-s − 4·59-s + 4.74·61-s + 65-s − 3.48·67-s + 11.4·71-s − 3.83·73-s + 6.41·77-s + 12.9·79-s − 4·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.92·7-s − 0.379·11-s − 0.277·13-s + 1.72·17-s − 1.45·19-s + 1.17·23-s + 0.200·25-s + 0.808·29-s + 0.359·31-s + 0.861·35-s + 0.121·37-s − 0.115·41-s − 0.836·43-s + 0.799·47-s + 2.70·49-s − 0.925·53-s + 0.169·55-s − 0.520·59-s + 0.607·61-s + 0.124·65-s − 0.425·67-s + 1.35·71-s − 0.449·73-s + 0.731·77-s + 1.45·79-s − 0.439·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + 5.09T + 7T^{2} \) |
| 11 | \( 1 + 1.25T + 11T^{2} \) |
| 17 | \( 1 - 7.09T + 17T^{2} \) |
| 19 | \( 1 + 6.35T + 19T^{2} \) |
| 23 | \( 1 - 5.61T + 23T^{2} \) |
| 29 | \( 1 - 4.35T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 - 0.741T + 37T^{2} \) |
| 41 | \( 1 + 0.741T + 41T^{2} \) |
| 43 | \( 1 + 5.48T + 43T^{2} \) |
| 47 | \( 1 - 5.48T + 47T^{2} \) |
| 53 | \( 1 + 6.74T + 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 - 4.74T + 61T^{2} \) |
| 67 | \( 1 + 3.48T + 67T^{2} \) |
| 71 | \( 1 - 11.4T + 71T^{2} \) |
| 73 | \( 1 + 3.83T + 73T^{2} \) |
| 79 | \( 1 - 12.9T + 79T^{2} \) |
| 83 | \( 1 + 4T + 83T^{2} \) |
| 89 | \( 1 + 15.9T + 89T^{2} \) |
| 97 | \( 1 - 13.0T + 97T^{2} \) |
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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
−7.26112453167588144625845053876, −6.67222567108977010451146128222, −6.13984960424183795399203353157, −5.36598732205704319889061615519, −4.53548134399020528820530213920, −3.62144635270481814142897677613, −3.13153889193863127122533121610, −2.45784042759682356044135254918, −0.977472791958177563371237569422, 0,
0.977472791958177563371237569422, 2.45784042759682356044135254918, 3.13153889193863127122533121610, 3.62144635270481814142897677613, 4.53548134399020528820530213920, 5.36598732205704319889061615519, 6.13984960424183795399203353157, 6.67222567108977010451146128222, 7.26112453167588144625845053876
