| L(s) = 1 | + 3.07·3-s + 0.386·5-s − 0.194·7-s + 6.43·9-s − 2.36·11-s − 1.22·13-s + 1.18·15-s − 5.04·17-s − 4.24·19-s − 0.598·21-s − 1.53·23-s − 4.85·25-s + 10.5·27-s − 7.31·29-s − 3.33·31-s − 7.26·33-s − 0.0752·35-s − 2.17·37-s − 3.76·39-s − 1.04·41-s + 1.53·43-s + 2.48·45-s + 1.08·47-s − 6.96·49-s − 15.5·51-s + 1.55·53-s − 0.912·55-s + ⋯ |
| L(s) = 1 | + 1.77·3-s + 0.172·5-s − 0.0736·7-s + 2.14·9-s − 0.712·11-s − 0.340·13-s + 0.306·15-s − 1.22·17-s − 0.973·19-s − 0.130·21-s − 0.319·23-s − 0.970·25-s + 2.03·27-s − 1.35·29-s − 0.599·31-s − 1.26·33-s − 0.0127·35-s − 0.356·37-s − 0.603·39-s − 0.163·41-s + 0.233·43-s + 0.370·45-s + 0.158·47-s − 0.994·49-s − 2.17·51-s + 0.213·53-s − 0.123·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 503 | \( 1 - T \) |
| good | 3 | \( 1 - 3.07T + 3T^{2} \) |
| 5 | \( 1 - 0.386T + 5T^{2} \) |
| 7 | \( 1 + 0.194T + 7T^{2} \) |
| 11 | \( 1 + 2.36T + 11T^{2} \) |
| 13 | \( 1 + 1.22T + 13T^{2} \) |
| 17 | \( 1 + 5.04T + 17T^{2} \) |
| 19 | \( 1 + 4.24T + 19T^{2} \) |
| 23 | \( 1 + 1.53T + 23T^{2} \) |
| 29 | \( 1 + 7.31T + 29T^{2} \) |
| 31 | \( 1 + 3.33T + 31T^{2} \) |
| 37 | \( 1 + 2.17T + 37T^{2} \) |
| 41 | \( 1 + 1.04T + 41T^{2} \) |
| 43 | \( 1 - 1.53T + 43T^{2} \) |
| 47 | \( 1 - 1.08T + 47T^{2} \) |
| 53 | \( 1 - 1.55T + 53T^{2} \) |
| 59 | \( 1 + 14.8T + 59T^{2} \) |
| 61 | \( 1 + 5.45T + 61T^{2} \) |
| 67 | \( 1 - 11.3T + 67T^{2} \) |
| 71 | \( 1 - 11.4T + 71T^{2} \) |
| 73 | \( 1 + 0.902T + 73T^{2} \) |
| 79 | \( 1 - 12.7T + 79T^{2} \) |
| 83 | \( 1 + 6.10T + 83T^{2} \) |
| 89 | \( 1 - 6.44T + 89T^{2} \) |
| 97 | \( 1 + 12.9T + 97T^{2} \) |
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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
−7.73848767959801849284298789791, −6.99479141136372904470870036725, −6.27890971009660680523379852497, −5.27550471745736313642176598754, −4.39074272623757371323332052234, −3.80995116944800720540927464681, −3.02665180558042759351881926450, −2.12452960313394461885542522535, −1.89193302942981337190482801975, 0,
1.89193302942981337190482801975, 2.12452960313394461885542522535, 3.02665180558042759351881926450, 3.80995116944800720540927464681, 4.39074272623757371323332052234, 5.27550471745736313642176598754, 6.27890971009660680523379852497, 6.99479141136372904470870036725, 7.73848767959801849284298789791
