| L(s) = 1 | − i·2-s + 3-s − 4-s + i·5-s − i·6-s − 0.307i·7-s + i·8-s + 9-s + 10-s − 0.335i·11-s − 12-s − 0.307·14-s + i·15-s + 16-s + 6.85·17-s − i·18-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 0.577·3-s − 0.5·4-s + 0.447i·5-s − 0.408i·6-s − 0.116i·7-s + 0.353i·8-s + 0.333·9-s + 0.316·10-s − 0.101i·11-s − 0.288·12-s − 0.0823·14-s + 0.258i·15-s + 0.250·16-s + 1.66·17-s − 0.235i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0304i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0304i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.262545664\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.262545664\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - iT \) |
| 13 | \( 1 \) |
| good | 7 | \( 1 + 0.307iT - 7T^{2} \) |
| 11 | \( 1 + 0.335iT - 11T^{2} \) |
| 17 | \( 1 - 6.85T + 17T^{2} \) |
| 19 | \( 1 - 5.80iT - 19T^{2} \) |
| 23 | \( 1 + 2.35T + 23T^{2} \) |
| 29 | \( 1 + 5.91T + 29T^{2} \) |
| 31 | \( 1 - 0.0609iT - 31T^{2} \) |
| 37 | \( 1 + 7.07iT - 37T^{2} \) |
| 41 | \( 1 - 3.40iT - 41T^{2} \) |
| 43 | \( 1 - 0.0489T + 43T^{2} \) |
| 47 | \( 1 + 7iT - 47T^{2} \) |
| 53 | \( 1 - 12.1T + 53T^{2} \) |
| 59 | \( 1 - 13.1iT - 59T^{2} \) |
| 61 | \( 1 - 2.14T + 61T^{2} \) |
| 67 | \( 1 - 11.9iT - 67T^{2} \) |
| 71 | \( 1 - 9.56iT - 71T^{2} \) |
| 73 | \( 1 + 2.31iT - 73T^{2} \) |
| 79 | \( 1 + 0.0760T + 79T^{2} \) |
| 83 | \( 1 - 3.84iT - 83T^{2} \) |
| 89 | \( 1 - 4.41iT - 89T^{2} \) |
| 97 | \( 1 - 6.13iT - 97T^{2} \) |
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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
−8.255861318015385814912143857950, −7.62503138114425196402435895554, −7.04908881638200399132951771786, −5.75164473009318656483606279957, −5.50528714012578041608384947995, −3.98902694939015341084779198709, −3.79727425631430690649028525444, −2.84500305633886341567744098985, −2.01326901708410691471859695820, −1.02989473914475097512204666517,
0.65979283773728119263811915740, 1.84838658611531074187978966018, 2.99882006160799491865799085244, 3.77229790293537996765107911806, 4.65957924410948527469989431783, 5.33848124796563601522821789099, 6.03096483864702229009591333145, 6.95849237578804001842320719169, 7.58007426986017066158944983103, 8.141835332983237138582015810955
