| L(s) = 1 | + 2.91·3-s + 2.60·5-s + 3.20·7-s + 5.52·9-s + 0.803·11-s + 6.10·13-s + 7.60·15-s − 1.52·17-s − 4.85·19-s + 9.36·21-s − 4.53·23-s + 1.79·25-s + 7.37·27-s − 7.34·29-s − 4.76·31-s + 2.34·33-s + 8.36·35-s + 0.000294·37-s + 17.8·39-s + 6.43·41-s + 12.3·43-s + 14.3·45-s − 3.72·47-s + 3.29·49-s − 4.45·51-s − 0.462·53-s + 2.09·55-s + ⋯ |
| L(s) = 1 | + 1.68·3-s + 1.16·5-s + 1.21·7-s + 1.84·9-s + 0.242·11-s + 1.69·13-s + 1.96·15-s − 0.369·17-s − 1.11·19-s + 2.04·21-s − 0.945·23-s + 0.358·25-s + 1.41·27-s − 1.36·29-s − 0.855·31-s + 0.408·33-s + 1.41·35-s + 4.84e−5·37-s + 2.85·39-s + 1.00·41-s + 1.88·43-s + 2.14·45-s − 0.543·47-s + 0.470·49-s − 0.623·51-s − 0.0635·53-s + 0.282·55-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
| \(L(1)\) |
\(\approx\) |
\(6.315046310\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.315046310\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.91T + 3T^{2} \) |
| 5 | \( 1 - 2.60T + 5T^{2} \) |
| 7 | \( 1 - 3.20T + 7T^{2} \) |
| 11 | \( 1 - 0.803T + 11T^{2} \) |
| 13 | \( 1 - 6.10T + 13T^{2} \) |
| 17 | \( 1 + 1.52T + 17T^{2} \) |
| 19 | \( 1 + 4.85T + 19T^{2} \) |
| 23 | \( 1 + 4.53T + 23T^{2} \) |
| 29 | \( 1 + 7.34T + 29T^{2} \) |
| 31 | \( 1 + 4.76T + 31T^{2} \) |
| 37 | \( 1 - 0.000294T + 37T^{2} \) |
| 41 | \( 1 - 6.43T + 41T^{2} \) |
| 43 | \( 1 - 12.3T + 43T^{2} \) |
| 47 | \( 1 + 3.72T + 47T^{2} \) |
| 53 | \( 1 + 0.462T + 53T^{2} \) |
| 59 | \( 1 + 7.89T + 59T^{2} \) |
| 61 | \( 1 + 4.63T + 61T^{2} \) |
| 67 | \( 1 + 0.890T + 67T^{2} \) |
| 71 | \( 1 + 1.96T + 71T^{2} \) |
| 73 | \( 1 - 10.8T + 73T^{2} \) |
| 79 | \( 1 - 8.42T + 79T^{2} \) |
| 83 | \( 1 - 14.5T + 83T^{2} \) |
| 89 | \( 1 + 0.680T + 89T^{2} \) |
| 97 | \( 1 - 5.67T + 97T^{2} \) |
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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
−7.86925270170500972166833102516, −7.52450471041151501427672092207, −6.27407865857527234789290395512, −5.98473086767585203972190785657, −4.89497945457718993106225298779, −4.00353142697651936896042638746, −3.61957910533267541919276138960, −2.33576941555391340434949373422, −1.97809822266890488371828616214, −1.32695098229557957017945400206,
1.32695098229557957017945400206, 1.97809822266890488371828616214, 2.33576941555391340434949373422, 3.61957910533267541919276138960, 4.00353142697651936896042638746, 4.89497945457718993106225298779, 5.98473086767585203972190785657, 6.27407865857527234789290395512, 7.52450471041151501427672092207, 7.86925270170500972166833102516
