| L(s) = 1 | − 3-s + 3.05·7-s + 9-s − 3.65·11-s − 5.70·13-s + 6.70·17-s + 2.31·19-s − 3.05·21-s − 7.75·23-s − 27-s + 4.18·29-s + 31-s + 3.65·33-s + 2.91·37-s + 5.70·39-s − 12.2·41-s + 9.31·43-s − 3.31·47-s + 2.31·49-s − 6.70·51-s − 3.91·53-s − 2.31·57-s + 7.86·59-s + 2.10·61-s + 3.05·63-s + 14.5·67-s + 7.75·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.15·7-s + 0.333·9-s − 1.10·11-s − 1.58·13-s + 1.62·17-s + 0.530·19-s − 0.665·21-s − 1.61·23-s − 0.192·27-s + 0.777·29-s + 0.179·31-s + 0.636·33-s + 0.479·37-s + 0.913·39-s − 1.91·41-s + 1.41·43-s − 0.482·47-s + 0.330·49-s − 0.939·51-s − 0.537·53-s − 0.306·57-s + 1.02·59-s + 0.269·61-s + 0.384·63-s + 1.78·67-s + 0.933·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 31 | \( 1 - T \) |
| good | 7 | \( 1 - 3.05T + 7T^{2} \) |
| 11 | \( 1 + 3.65T + 11T^{2} \) |
| 13 | \( 1 + 5.70T + 13T^{2} \) |
| 17 | \( 1 - 6.70T + 17T^{2} \) |
| 19 | \( 1 - 2.31T + 19T^{2} \) |
| 23 | \( 1 + 7.75T + 23T^{2} \) |
| 29 | \( 1 - 4.18T + 29T^{2} \) |
| 37 | \( 1 - 2.91T + 37T^{2} \) |
| 41 | \( 1 + 12.2T + 41T^{2} \) |
| 43 | \( 1 - 9.31T + 43T^{2} \) |
| 47 | \( 1 + 3.31T + 47T^{2} \) |
| 53 | \( 1 + 3.91T + 53T^{2} \) |
| 59 | \( 1 - 7.86T + 59T^{2} \) |
| 61 | \( 1 - 2.10T + 61T^{2} \) |
| 67 | \( 1 - 14.5T + 67T^{2} \) |
| 71 | \( 1 - 4.10T + 71T^{2} \) |
| 73 | \( 1 - 4.79T + 73T^{2} \) |
| 79 | \( 1 + 13.5T + 79T^{2} \) |
| 83 | \( 1 + 10.4T + 83T^{2} \) |
| 89 | \( 1 + 13.7T + 89T^{2} \) |
| 97 | \( 1 + 16.9T + 97T^{2} \) |
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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
−7.47599194734342963771651873914, −6.79822957351552768413705694422, −5.70371458248687966894143766642, −5.30779501046563223477825656852, −4.81806876255509662594645141165, −4.00673111597917875836809944326, −2.90851405399184208306949236781, −2.15856600204699639950180475872, −1.18856302618762972219282118363, 0,
1.18856302618762972219282118363, 2.15856600204699639950180475872, 2.90851405399184208306949236781, 4.00673111597917875836809944326, 4.81806876255509662594645141165, 5.30779501046563223477825656852, 5.70371458248687966894143766642, 6.79822957351552768413705694422, 7.47599194734342963771651873914
