| L(s) = 1 | − 0.0212·3-s − 0.474·5-s + 0.949·7-s − 2.99·9-s − 5.65·11-s − 4.21·13-s + 0.0100·15-s + 1.35·17-s − 5.79·19-s − 0.0201·21-s + 7.38·23-s − 4.77·25-s + 0.127·27-s − 2.48·29-s + 5.09·31-s + 0.120·33-s − 0.450·35-s + 8.76·37-s + 0.0894·39-s − 0.109·41-s − 10.9·43-s + 1.42·45-s + 9.85·47-s − 6.09·49-s − 0.0287·51-s − 11.4·53-s + 2.68·55-s + ⋯ |
| L(s) = 1 | − 0.0122·3-s − 0.212·5-s + 0.358·7-s − 0.999·9-s − 1.70·11-s − 1.16·13-s + 0.00260·15-s + 0.328·17-s − 1.32·19-s − 0.00439·21-s + 1.54·23-s − 0.954·25-s + 0.0245·27-s − 0.461·29-s + 0.915·31-s + 0.0208·33-s − 0.0761·35-s + 1.44·37-s + 0.0143·39-s − 0.0170·41-s − 1.67·43-s + 0.212·45-s + 1.43·47-s − 0.871·49-s − 0.00403·51-s − 1.56·53-s + 0.361·55-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
| \(L(1)\) |
\(\approx\) |
\(0.7627462563\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7627462563\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.0212T + 3T^{2} \) |
| 5 | \( 1 + 0.474T + 5T^{2} \) |
| 7 | \( 1 - 0.949T + 7T^{2} \) |
| 11 | \( 1 + 5.65T + 11T^{2} \) |
| 13 | \( 1 + 4.21T + 13T^{2} \) |
| 17 | \( 1 - 1.35T + 17T^{2} \) |
| 19 | \( 1 + 5.79T + 19T^{2} \) |
| 23 | \( 1 - 7.38T + 23T^{2} \) |
| 29 | \( 1 + 2.48T + 29T^{2} \) |
| 31 | \( 1 - 5.09T + 31T^{2} \) |
| 37 | \( 1 - 8.76T + 37T^{2} \) |
| 41 | \( 1 + 0.109T + 41T^{2} \) |
| 43 | \( 1 + 10.9T + 43T^{2} \) |
| 47 | \( 1 - 9.85T + 47T^{2} \) |
| 53 | \( 1 + 11.4T + 53T^{2} \) |
| 59 | \( 1 + 9.75T + 59T^{2} \) |
| 61 | \( 1 + 13.1T + 61T^{2} \) |
| 67 | \( 1 - 6.25T + 67T^{2} \) |
| 71 | \( 1 - 12.2T + 71T^{2} \) |
| 73 | \( 1 + 15.1T + 73T^{2} \) |
| 79 | \( 1 - 11.4T + 79T^{2} \) |
| 83 | \( 1 - 1.29T + 83T^{2} \) |
| 89 | \( 1 + 0.406T + 89T^{2} \) |
| 97 | \( 1 - 10.1T + 97T^{2} \) |
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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
−7.955848561205731062799507380371, −7.33511937699860899798987753797, −6.36433234302025787521562637787, −5.71428254066673252670260441605, −4.87318084231103100806259316443, −4.63618599109911234886374035149, −3.26408957420878393754033763564, −2.70968802130938848310862030227, −1.97396728649223778284131922224, −0.40244869992200476822667049273,
0.40244869992200476822667049273, 1.97396728649223778284131922224, 2.70968802130938848310862030227, 3.26408957420878393754033763564, 4.63618599109911234886374035149, 4.87318084231103100806259316443, 5.71428254066673252670260441605, 6.36433234302025787521562637787, 7.33511937699860899798987753797, 7.955848561205731062799507380371
