| L(s) = 1 | − 1.43·2-s + 0.0731·4-s + 0.926·5-s + 2.77·8-s − 1.33·10-s − 4.21·11-s − 13-s − 4.14·16-s − 2.87·17-s + 1.28·19-s + 0.0678·20-s + 6.06·22-s + 8.02·23-s − 4.14·25-s + 1.43·26-s − 3.28·29-s + 7.04·31-s + 0.413·32-s + 4.14·34-s + 8.57·37-s − 1.85·38-s + 2.57·40-s − 12.0·41-s + 7.14·43-s − 0.308·44-s − 11.5·46-s + 1.95·47-s + ⋯ |
| L(s) = 1 | − 1.01·2-s + 0.0365·4-s + 0.414·5-s + 0.980·8-s − 0.422·10-s − 1.27·11-s − 0.277·13-s − 1.03·16-s − 0.698·17-s + 0.295·19-s + 0.0151·20-s + 1.29·22-s + 1.67·23-s − 0.828·25-s + 0.282·26-s − 0.610·29-s + 1.26·31-s + 0.0731·32-s + 0.711·34-s + 1.40·37-s − 0.300·38-s + 0.406·40-s − 1.88·41-s + 1.08·43-s − 0.0464·44-s − 1.70·46-s + 0.284·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5733 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5733 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 + 1.43T + 2T^{2} \) |
| 5 | \( 1 - 0.926T + 5T^{2} \) |
| 11 | \( 1 + 4.21T + 11T^{2} \) |
| 17 | \( 1 + 2.87T + 17T^{2} \) |
| 19 | \( 1 - 1.28T + 19T^{2} \) |
| 23 | \( 1 - 8.02T + 23T^{2} \) |
| 29 | \( 1 + 3.28T + 29T^{2} \) |
| 31 | \( 1 - 7.04T + 31T^{2} \) |
| 37 | \( 1 - 8.57T + 37T^{2} \) |
| 41 | \( 1 + 12.0T + 41T^{2} \) |
| 43 | \( 1 - 7.14T + 43T^{2} \) |
| 47 | \( 1 - 1.95T + 47T^{2} \) |
| 53 | \( 1 - 5.14T + 53T^{2} \) |
| 59 | \( 1 + 7.33T + 59T^{2} \) |
| 61 | \( 1 + 7.75T + 61T^{2} \) |
| 67 | \( 1 - 12.0T + 67T^{2} \) |
| 71 | \( 1 + 10.7T + 71T^{2} \) |
| 73 | \( 1 - 8.32T + 73T^{2} \) |
| 79 | \( 1 - 4.47T + 79T^{2} \) |
| 83 | \( 1 + 3.80T + 83T^{2} \) |
| 89 | \( 1 + 5.64T + 89T^{2} \) |
| 97 | \( 1 + 6.90T + 97T^{2} \) |
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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
−7.88022923768521512617289112024, −7.30012484176485825628825041203, −6.54927108246297572888901124092, −5.53795868160919085390447662320, −4.93839525140265462414638901037, −4.20542055827670338463232149552, −2.93962049313617378947003947423, −2.21237405843254983204492030968, −1.11834739631990087472559495899, 0,
1.11834739631990087472559495899, 2.21237405843254983204492030968, 2.93962049313617378947003947423, 4.20542055827670338463232149552, 4.93839525140265462414638901037, 5.53795868160919085390447662320, 6.54927108246297572888901124092, 7.30012484176485825628825041203, 7.88022923768521512617289112024
