| L(s) = 1 | + 2.82i·2-s + 5.19·3-s − 8.00·4-s + 26.1·5-s + 14.6i·6-s − 51.7·7-s − 22.6i·8-s + 27·9-s + 73.9i·10-s + 34.2i·11-s − 41.5·12-s − 269. i·13-s − 146. i·14-s + 135.·15-s + 64.0·16-s + 472.·17-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + 0.577·3-s − 0.500·4-s + 1.04·5-s + 0.408i·6-s − 1.05·7-s − 0.353i·8-s + 0.333·9-s + 0.739i·10-s + 0.283i·11-s − 0.288·12-s − 1.59i·13-s − 0.746i·14-s + 0.603·15-s + 0.250·16-s + 1.63·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.871 - 0.490i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.871 - 0.490i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(2.683578309\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.683578309\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2.82iT \) |
| 3 | \( 1 - 5.19T \) |
| 59 | \( 1 + (-1.70e3 - 3.03e3i)T \) |
| good | 5 | \( 1 - 26.1T + 625T^{2} \) |
| 7 | \( 1 + 51.7T + 2.40e3T^{2} \) |
| 11 | \( 1 - 34.2iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 269. iT - 2.85e4T^{2} \) |
| 17 | \( 1 - 472.T + 8.35e4T^{2} \) |
| 19 | \( 1 - 44.0T + 1.30e5T^{2} \) |
| 23 | \( 1 - 650. iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 1.25e3T + 7.07e5T^{2} \) |
| 31 | \( 1 + 441. iT - 9.23e5T^{2} \) |
| 37 | \( 1 + 801. iT - 1.87e6T^{2} \) |
| 41 | \( 1 - 2.40e3T + 2.82e6T^{2} \) |
| 43 | \( 1 + 3.07e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 - 785. iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 4.18e3T + 7.89e6T^{2} \) |
| 61 | \( 1 - 322. iT - 1.38e7T^{2} \) |
| 67 | \( 1 - 392. iT - 2.01e7T^{2} \) |
| 71 | \( 1 + 1.96e3T + 2.54e7T^{2} \) |
| 73 | \( 1 - 445. iT - 2.83e7T^{2} \) |
| 79 | \( 1 - 2.73e3T + 3.89e7T^{2} \) |
| 83 | \( 1 - 9.31e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 2.93e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 2.29e3iT - 8.85e7T^{2} \) |
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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
−10.29117727617885993429883927807, −9.945353270809104559440864000760, −9.135792230710318821038671404779, −7.976643789360916077474825019458, −7.18979973226597185834038898355, −5.91653373002188366943124578765, −5.44015975214849750678881205750, −3.68123791325882255045945211426, −2.68234911205437327851725585123, −0.910245657094894232773330070845,
1.09395068044727780266339966529, 2.37014124323642579594610165712, 3.29743811235361746086795687246, 4.55093639200768289744678123289, 5.96478730583101212893693039045, 6.79088825257412446374312066229, 8.253145501773135511100834088576, 9.267189054013420944080861322221, 9.782600490729835441724306055645, 10.46516783061145144488651444873
