| L(s) = 1 | + 3.30·3-s − 2.55·7-s + 7.93·9-s + 2.72·11-s − 7.12·13-s − 0.924·17-s − 7.51·19-s − 8.43·21-s − 23-s + 16.3·27-s − 2.38·29-s − 0.866·31-s + 9.00·33-s − 0.352·37-s − 23.5·39-s + 4.34·41-s − 13.3·47-s − 0.495·49-s − 3.05·51-s − 3.99·53-s − 24.8·57-s + 3.84·59-s − 9.14·61-s − 20.2·63-s − 3.15·67-s − 3.30·69-s + 6.07·71-s + ⋯ |
| L(s) = 1 | + 1.90·3-s − 0.963·7-s + 2.64·9-s + 0.821·11-s − 1.97·13-s − 0.224·17-s − 1.72·19-s − 1.84·21-s − 0.208·23-s + 3.13·27-s − 0.442·29-s − 0.155·31-s + 1.56·33-s − 0.0580·37-s − 3.77·39-s + 0.677·41-s − 1.94·47-s − 0.0707·49-s − 0.427·51-s − 0.548·53-s − 3.29·57-s + 0.500·59-s − 1.17·61-s − 2.54·63-s − 0.385·67-s − 0.398·69-s + 0.721·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 - 3.30T + 3T^{2} \) |
| 7 | \( 1 + 2.55T + 7T^{2} \) |
| 11 | \( 1 - 2.72T + 11T^{2} \) |
| 13 | \( 1 + 7.12T + 13T^{2} \) |
| 17 | \( 1 + 0.924T + 17T^{2} \) |
| 19 | \( 1 + 7.51T + 19T^{2} \) |
| 29 | \( 1 + 2.38T + 29T^{2} \) |
| 31 | \( 1 + 0.866T + 31T^{2} \) |
| 37 | \( 1 + 0.352T + 37T^{2} \) |
| 41 | \( 1 - 4.34T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 13.3T + 47T^{2} \) |
| 53 | \( 1 + 3.99T + 53T^{2} \) |
| 59 | \( 1 - 3.84T + 59T^{2} \) |
| 61 | \( 1 + 9.14T + 61T^{2} \) |
| 67 | \( 1 + 3.15T + 67T^{2} \) |
| 71 | \( 1 - 6.07T + 71T^{2} \) |
| 73 | \( 1 - 11.3T + 73T^{2} \) |
| 79 | \( 1 - 12.0T + 79T^{2} \) |
| 83 | \( 1 + 6.35T + 83T^{2} \) |
| 89 | \( 1 + 9.71T + 89T^{2} \) |
| 97 | \( 1 + 8.76T + 97T^{2} \) |
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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
−7.57202223860692592644599968425, −6.64708126450777252075287957152, −6.53681241591758445568896575022, −5.04732317402108454487813222308, −4.29075212767809678857583409335, −3.75735337424898802170193554830, −2.93138060591094635251267592160, −2.35998035149417267572294165892, −1.67060054343597940841336044208, 0,
1.67060054343597940841336044208, 2.35998035149417267572294165892, 2.93138060591094635251267592160, 3.75735337424898802170193554830, 4.29075212767809678857583409335, 5.04732317402108454487813222308, 6.53681241591758445568896575022, 6.64708126450777252075287957152, 7.57202223860692592644599968425
