| L(s) = 1 | + 2.05·3-s + 3.84·5-s + 4.00·7-s + 1.23·9-s − 3.28·11-s − 3.96·13-s + 7.92·15-s + 0.871·17-s + 3.06·19-s + 8.23·21-s + 1.95·23-s + 9.82·25-s − 3.62·27-s + 0.855·29-s + 6.85·31-s − 6.75·33-s + 15.4·35-s − 1.69·37-s − 8.15·39-s + 6.10·41-s − 2.26·43-s + 4.76·45-s − 3.98·47-s + 9.00·49-s + 1.79·51-s − 0.873·53-s − 12.6·55-s + ⋯ |
| L(s) = 1 | + 1.18·3-s + 1.72·5-s + 1.51·7-s + 0.412·9-s − 0.989·11-s − 1.09·13-s + 2.04·15-s + 0.211·17-s + 0.702·19-s + 1.79·21-s + 0.406·23-s + 1.96·25-s − 0.698·27-s + 0.158·29-s + 1.23·31-s − 1.17·33-s + 2.60·35-s − 0.279·37-s − 1.30·39-s + 0.953·41-s − 0.345·43-s + 0.710·45-s − 0.580·47-s + 1.28·49-s + 0.251·51-s − 0.120·53-s − 1.70·55-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
| \(L(1)\) |
\(\approx\) |
\(5.268726617\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.268726617\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.05T + 3T^{2} \) |
| 5 | \( 1 - 3.84T + 5T^{2} \) |
| 7 | \( 1 - 4.00T + 7T^{2} \) |
| 11 | \( 1 + 3.28T + 11T^{2} \) |
| 13 | \( 1 + 3.96T + 13T^{2} \) |
| 17 | \( 1 - 0.871T + 17T^{2} \) |
| 19 | \( 1 - 3.06T + 19T^{2} \) |
| 23 | \( 1 - 1.95T + 23T^{2} \) |
| 29 | \( 1 - 0.855T + 29T^{2} \) |
| 31 | \( 1 - 6.85T + 31T^{2} \) |
| 37 | \( 1 + 1.69T + 37T^{2} \) |
| 41 | \( 1 - 6.10T + 41T^{2} \) |
| 43 | \( 1 + 2.26T + 43T^{2} \) |
| 47 | \( 1 + 3.98T + 47T^{2} \) |
| 53 | \( 1 + 0.873T + 53T^{2} \) |
| 59 | \( 1 + 6.66T + 59T^{2} \) |
| 61 | \( 1 - 12.1T + 61T^{2} \) |
| 67 | \( 1 - 15.5T + 67T^{2} \) |
| 71 | \( 1 + 6.93T + 71T^{2} \) |
| 73 | \( 1 - 3.14T + 73T^{2} \) |
| 79 | \( 1 - 0.0135T + 79T^{2} \) |
| 83 | \( 1 - 2.36T + 83T^{2} \) |
| 89 | \( 1 - 12.8T + 89T^{2} \) |
| 97 | \( 1 + 7.16T + 97T^{2} \) |
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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
−7.972675210424736653744000823693, −7.36593776689028029016978990338, −6.49256226723258981636838739385, −5.40019598999484676640787477809, −5.23185029707794987825278160454, −4.45431272955181764446387770394, −3.13846747138238083250249438811, −2.45863633322491978680433975861, −2.08061915683364313090893892706, −1.14757533318632374357764737623,
1.14757533318632374357764737623, 2.08061915683364313090893892706, 2.45863633322491978680433975861, 3.13846747138238083250249438811, 4.45431272955181764446387770394, 5.23185029707794987825278160454, 5.40019598999484676640787477809, 6.49256226723258981636838739385, 7.36593776689028029016978990338, 7.972675210424736653744000823693
