| 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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.44T + 2T^{2} \) |
| 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
−7.43641051499140997212587974607, −6.89908146621326582435040019995, −6.10869383766995218639439079749, −5.52551420064136491400203228167, −4.68867824260123531674919260920, −4.23646699986043749923897263231, −3.42872481169032329473625400646, −2.58534236542313206521991202055, −1.91205652640730474214951925095, 0,
1.91205652640730474214951925095, 2.58534236542313206521991202055, 3.42872481169032329473625400646, 4.23646699986043749923897263231, 4.68867824260123531674919260920, 5.52551420064136491400203228167, 6.10869383766995218639439079749, 6.89908146621326582435040019995, 7.43641051499140997212587974607
