| L(s) = 1 | + 5-s + 1.19·7-s − 1.19·11-s + 13-s − 6.17·17-s − 6.97·19-s + 4.17·23-s + 25-s − 6·29-s + 2.97·31-s + 1.19·35-s + 7.78·37-s + 6.17·41-s + 9.95·43-s − 1.02·47-s − 5.56·49-s − 10.1·53-s − 1.19·55-s + 5.37·59-s + 12.5·61-s + 65-s − 9.37·67-s − 5.19·71-s − 11.9·73-s − 1.43·77-s + 1.78·79-s − 5.37·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.452·7-s − 0.360·11-s + 0.277·13-s − 1.49·17-s − 1.60·19-s + 0.870·23-s + 0.200·25-s − 1.11·29-s + 0.534·31-s + 0.202·35-s + 1.27·37-s + 0.964·41-s + 1.51·43-s − 0.148·47-s − 0.795·49-s − 1.39·53-s − 0.161·55-s + 0.699·59-s + 1.60·61-s + 0.124·65-s − 1.14·67-s − 0.616·71-s − 1.39·73-s − 0.163·77-s + 0.200·79-s − 0.589·83-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 | \( 1 - 1.19T + 7T^{2} \) |
| 11 | \( 1 + 1.19T + 11T^{2} \) |
| 17 | \( 1 + 6.17T + 17T^{2} \) |
| 19 | \( 1 + 6.97T + 19T^{2} \) |
| 23 | \( 1 - 4.17T + 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 - 2.97T + 31T^{2} \) |
| 37 | \( 1 - 7.78T + 37T^{2} \) |
| 41 | \( 1 - 6.17T + 41T^{2} \) |
| 43 | \( 1 - 9.95T + 43T^{2} \) |
| 47 | \( 1 + 1.02T + 47T^{2} \) |
| 53 | \( 1 + 10.1T + 53T^{2} \) |
| 59 | \( 1 - 5.37T + 59T^{2} \) |
| 61 | \( 1 - 12.5T + 61T^{2} \) |
| 67 | \( 1 + 9.37T + 67T^{2} \) |
| 71 | \( 1 + 5.19T + 71T^{2} \) |
| 73 | \( 1 + 11.9T + 73T^{2} \) |
| 79 | \( 1 - 1.78T + 79T^{2} \) |
| 83 | \( 1 + 5.37T + 83T^{2} \) |
| 89 | \( 1 + 10.1T + 89T^{2} \) |
| 97 | \( 1 + 1.82T + 97T^{2} \) |
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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
−7.36904090763144214058670528283, −6.57669499194461959635170036677, −6.09024610070575552062979384903, −5.32149414825110179596937908779, −4.44427261807916518284124690327, −4.11296974696803624852310250561, −2.78150704409632401017580954555, −2.26598433142113130665580408162, −1.32410764524985695962530822765, 0,
1.32410764524985695962530822765, 2.26598433142113130665580408162, 2.78150704409632401017580954555, 4.11296974696803624852310250561, 4.44427261807916518284124690327, 5.32149414825110179596937908779, 6.09024610070575552062979384903, 6.57669499194461959635170036677, 7.36904090763144214058670528283
