| L(s) = 1 | + 2.70·3-s + 2.99·5-s + 2.81·7-s + 4.34·9-s + 3.12·11-s + 1.74·13-s + 8.11·15-s + 0.354·17-s + 0.768·19-s + 7.61·21-s − 2.62·23-s + 3.97·25-s + 3.63·27-s − 2.66·29-s − 7.34·31-s + 8.46·33-s + 8.41·35-s − 5.78·37-s + 4.71·39-s − 6.53·41-s − 11.2·43-s + 13.0·45-s + 11.5·47-s + 0.897·49-s + 0.960·51-s + 5.59·53-s + 9.35·55-s + ⋯ |
| L(s) = 1 | + 1.56·3-s + 1.33·5-s + 1.06·7-s + 1.44·9-s + 0.941·11-s + 0.482·13-s + 2.09·15-s + 0.0859·17-s + 0.176·19-s + 1.66·21-s − 0.547·23-s + 0.794·25-s + 0.700·27-s − 0.494·29-s − 1.31·31-s + 1.47·33-s + 1.42·35-s − 0.950·37-s + 0.755·39-s − 1.02·41-s − 1.70·43-s + 1.93·45-s + 1.68·47-s + 0.128·49-s + 0.134·51-s + 0.768·53-s + 1.26·55-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
| \(L(1)\) |
\(\approx\) |
\(6.220485084\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.220485084\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.70T + 3T^{2} \) |
| 5 | \( 1 - 2.99T + 5T^{2} \) |
| 7 | \( 1 - 2.81T + 7T^{2} \) |
| 11 | \( 1 - 3.12T + 11T^{2} \) |
| 13 | \( 1 - 1.74T + 13T^{2} \) |
| 17 | \( 1 - 0.354T + 17T^{2} \) |
| 19 | \( 1 - 0.768T + 19T^{2} \) |
| 23 | \( 1 + 2.62T + 23T^{2} \) |
| 29 | \( 1 + 2.66T + 29T^{2} \) |
| 31 | \( 1 + 7.34T + 31T^{2} \) |
| 37 | \( 1 + 5.78T + 37T^{2} \) |
| 41 | \( 1 + 6.53T + 41T^{2} \) |
| 43 | \( 1 + 11.2T + 43T^{2} \) |
| 47 | \( 1 - 11.5T + 47T^{2} \) |
| 53 | \( 1 - 5.59T + 53T^{2} \) |
| 59 | \( 1 - 5.94T + 59T^{2} \) |
| 61 | \( 1 - 13.2T + 61T^{2} \) |
| 67 | \( 1 - 14.9T + 67T^{2} \) |
| 71 | \( 1 - 1.16T + 71T^{2} \) |
| 73 | \( 1 - 7.40T + 73T^{2} \) |
| 79 | \( 1 + 1.11T + 79T^{2} \) |
| 83 | \( 1 + 12.0T + 83T^{2} \) |
| 89 | \( 1 - 8.47T + 89T^{2} \) |
| 97 | \( 1 + 11.3T + 97T^{2} \) |
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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
−8.071870262653461069357505039101, −7.16250819061330924562197488232, −6.64881059707847264218219629441, −5.59630259577213809570392273048, −5.15672044282384542975597681275, −3.93121048408955922015186501366, −3.61113560613739724816782438988, −2.41993081959219422072942130039, −1.87511028967338133020711390621, −1.34548048669101073432199872854,
1.34548048669101073432199872854, 1.87511028967338133020711390621, 2.41993081959219422072942130039, 3.61113560613739724816782438988, 3.93121048408955922015186501366, 5.15672044282384542975597681275, 5.59630259577213809570392273048, 6.64881059707847264218219629441, 7.16250819061330924562197488232, 8.071870262653461069357505039101
