| L(s) = 1 | + 2.44·2-s + 3.99·4-s + 0.910·5-s + 4.87·8-s + 2.22·10-s − 3.67·11-s − 13-s + 3.95·16-s + 7.18·17-s + 1.97·19-s + 3.63·20-s − 9.00·22-s + 0.596·23-s − 4.17·25-s − 2.44·26-s + 3.64·29-s + 7.08·31-s − 0.0786·32-s + 17.5·34-s + 0.710·37-s + 4.84·38-s + 4.43·40-s + 5.27·41-s + 11.0·43-s − 14.6·44-s + 1.46·46-s + 12.1·47-s + ⋯ |
| L(s) = 1 | + 1.73·2-s + 1.99·4-s + 0.407·5-s + 1.72·8-s + 0.704·10-s − 1.10·11-s − 0.277·13-s + 0.987·16-s + 1.74·17-s + 0.453·19-s + 0.812·20-s − 1.91·22-s + 0.124·23-s − 0.834·25-s − 0.480·26-s + 0.677·29-s + 1.27·31-s − 0.0139·32-s + 3.01·34-s + 0.116·37-s + 0.785·38-s + 0.701·40-s + 0.823·41-s + 1.68·43-s − 2.21·44-s + 0.215·46-s + 1.76·47-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
| \(L(1)\) |
\(\approx\) |
\(6.510829253\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.510829253\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.44T + 2T^{2} \) |
| 5 | \( 1 - 0.910T + 5T^{2} \) |
| 11 | \( 1 + 3.67T + 11T^{2} \) |
| 17 | \( 1 - 7.18T + 17T^{2} \) |
| 19 | \( 1 - 1.97T + 19T^{2} \) |
| 23 | \( 1 - 0.596T + 23T^{2} \) |
| 29 | \( 1 - 3.64T + 29T^{2} \) |
| 31 | \( 1 - 7.08T + 31T^{2} \) |
| 37 | \( 1 - 0.710T + 37T^{2} \) |
| 41 | \( 1 - 5.27T + 41T^{2} \) |
| 43 | \( 1 - 11.0T + 43T^{2} \) |
| 47 | \( 1 - 12.1T + 47T^{2} \) |
| 53 | \( 1 - 11.4T + 53T^{2} \) |
| 59 | \( 1 - 9.58T + 59T^{2} \) |
| 61 | \( 1 + 6.98T + 61T^{2} \) |
| 67 | \( 1 - 1.22T + 67T^{2} \) |
| 71 | \( 1 + 11.3T + 71T^{2} \) |
| 73 | \( 1 + 6.53T + 73T^{2} \) |
| 79 | \( 1 + 11.5T + 79T^{2} \) |
| 83 | \( 1 - 7.16T + 83T^{2} \) |
| 89 | \( 1 - 12.8T + 89T^{2} \) |
| 97 | \( 1 + 9.09T + 97T^{2} \) |
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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
−7.55672720972543977570327098006, −7.50611141007961290864308855772, −6.33199337022978923502086042326, −5.61763954418002696913496740009, −5.43029422686498172875242121096, −4.48267677594962619251184766592, −3.80712864267146896367255347370, −2.76088185299542786441067534025, −2.51122998091485589399661627935, −1.08979065815532064532752235635,
1.08979065815532064532752235635, 2.51122998091485589399661627935, 2.76088185299542786441067534025, 3.80712864267146896367255347370, 4.48267677594962619251184766592, 5.43029422686498172875242121096, 5.61763954418002696913496740009, 6.33199337022978923502086042326, 7.50611141007961290864308855772, 7.55672720972543977570327098006
