| L(s) = 1 | + 0.391·2-s − 1.84·4-s + 1.78·5-s − 1.50·8-s + 0.697·10-s − 2.39·11-s + 0.824·13-s + 3.10·16-s + 17-s + 2.47·19-s − 3.29·20-s − 0.936·22-s + 6.42·23-s − 1.81·25-s + 0.322·26-s − 0.325·29-s − 9.36·31-s + 4.22·32-s + 0.391·34-s + 5.73·37-s + 0.969·38-s − 2.68·40-s + 3.20·41-s + 5.52·43-s + 4.42·44-s + 2.51·46-s − 9.68·47-s + ⋯ |
| L(s) = 1 | + 0.276·2-s − 0.923·4-s + 0.797·5-s − 0.531·8-s + 0.220·10-s − 0.721·11-s + 0.228·13-s + 0.776·16-s + 0.242·17-s + 0.568·19-s − 0.736·20-s − 0.199·22-s + 1.33·23-s − 0.363·25-s + 0.0632·26-s − 0.0605·29-s − 1.68·31-s + 0.746·32-s + 0.0670·34-s + 0.943·37-s + 0.157·38-s − 0.424·40-s + 0.500·41-s + 0.842·43-s + 0.666·44-s + 0.370·46-s − 1.41·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7497 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7497 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.914714217\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.914714217\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 2 | \( 1 - 0.391T + 2T^{2} \) |
| 5 | \( 1 - 1.78T + 5T^{2} \) |
| 11 | \( 1 + 2.39T + 11T^{2} \) |
| 13 | \( 1 - 0.824T + 13T^{2} \) |
| 19 | \( 1 - 2.47T + 19T^{2} \) |
| 23 | \( 1 - 6.42T + 23T^{2} \) |
| 29 | \( 1 + 0.325T + 29T^{2} \) |
| 31 | \( 1 + 9.36T + 31T^{2} \) |
| 37 | \( 1 - 5.73T + 37T^{2} \) |
| 41 | \( 1 - 3.20T + 41T^{2} \) |
| 43 | \( 1 - 5.52T + 43T^{2} \) |
| 47 | \( 1 + 9.68T + 47T^{2} \) |
| 53 | \( 1 + 2.50T + 53T^{2} \) |
| 59 | \( 1 - 2.90T + 59T^{2} \) |
| 61 | \( 1 - 1.02T + 61T^{2} \) |
| 67 | \( 1 - 1.50T + 67T^{2} \) |
| 71 | \( 1 + 9.16T + 71T^{2} \) |
| 73 | \( 1 - 0.0705T + 73T^{2} \) |
| 79 | \( 1 + 0.730T + 79T^{2} \) |
| 83 | \( 1 - 5.53T + 83T^{2} \) |
| 89 | \( 1 - 0.313T + 89T^{2} \) |
| 97 | \( 1 - 7.79T + 97T^{2} \) |
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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
−7.85531147504640879604890418048, −7.30073921439477624752609685350, −6.26307062904952600798737821673, −5.62628189989118147832642161196, −5.18273990231873922393049745446, −4.44287681827831542712108004122, −3.51352782594213685692177009900, −2.85458742075463775811696678418, −1.77858455841253400704954081350, −0.67852125314346333424015179338,
0.67852125314346333424015179338, 1.77858455841253400704954081350, 2.85458742075463775811696678418, 3.51352782594213685692177009900, 4.44287681827831542712108004122, 5.18273990231873922393049745446, 5.62628189989118147832642161196, 6.26307062904952600798737821673, 7.30073921439477624752609685350, 7.85531147504640879604890418048
