| L(s) = 1 | + (0.952 − 2.93i)2-s + (0.594 + 0.432i)3-s + (18.1 + 13.2i)4-s + (28.1 − 48.2i)5-s + (1.83 − 1.33i)6-s + 57.2·7-s + (135. − 98.7i)8-s + (−74.9 − 230. i)9-s + (−114. − 128. i)10-s + (−24.1 + 74.2i)11-s + (5.11 + 15.7i)12-s + (210. + 646. i)13-s + (54.5 − 168. i)14-s + (37.6 − 16.5i)15-s + (62.2 + 191. i)16-s + (92.3 − 67.0i)17-s + ⋯ |
| L(s) = 1 | + (0.168 − 0.518i)2-s + (0.0381 + 0.0277i)3-s + (0.568 + 0.413i)4-s + (0.503 − 0.863i)5-s + (0.0208 − 0.0151i)6-s + 0.441·7-s + (0.750 − 0.545i)8-s + (−0.308 − 0.948i)9-s + (−0.362 − 0.406i)10-s + (−0.0600 + 0.184i)11-s + (0.0102 + 0.0315i)12-s + (0.344 + 1.06i)13-s + (0.0744 − 0.229i)14-s + (0.0431 − 0.0189i)15-s + (0.0608 + 0.187i)16-s + (0.0774 − 0.0562i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.693 + 0.720i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.693 + 0.720i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.80476 - 0.768365i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.80476 - 0.768365i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-28.1 + 48.2i)T \) |
| good | 2 | \( 1 + (-0.952 + 2.93i)T + (-25.8 - 18.8i)T^{2} \) |
| 3 | \( 1 + (-0.594 - 0.432i)T + (75.0 + 231. i)T^{2} \) |
| 7 | \( 1 - 57.2T + 1.68e4T^{2} \) |
| 11 | \( 1 + (24.1 - 74.2i)T + (-1.30e5 - 9.46e4i)T^{2} \) |
| 13 | \( 1 + (-210. - 646. i)T + (-3.00e5 + 2.18e5i)T^{2} \) |
| 17 | \( 1 + (-92.3 + 67.0i)T + (4.38e5 - 1.35e6i)T^{2} \) |
| 19 | \( 1 + (2.33e3 - 1.69e3i)T + (7.65e5 - 2.35e6i)T^{2} \) |
| 23 | \( 1 + (725. - 2.23e3i)T + (-5.20e6 - 3.78e6i)T^{2} \) |
| 29 | \( 1 + (4.40e3 + 3.19e3i)T + (6.33e6 + 1.95e7i)T^{2} \) |
| 31 | \( 1 + (-4.42e3 + 3.21e3i)T + (8.84e6 - 2.72e7i)T^{2} \) |
| 37 | \( 1 + (-3.11e3 - 9.58e3i)T + (-5.61e7 + 4.07e7i)T^{2} \) |
| 41 | \( 1 + (1.04e3 + 3.20e3i)T + (-9.37e7 + 6.80e7i)T^{2} \) |
| 43 | \( 1 + 1.58e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + (-1.08e4 - 7.87e3i)T + (7.08e7 + 2.18e8i)T^{2} \) |
| 53 | \( 1 + (-9.70e3 - 7.05e3i)T + (1.29e8 + 3.97e8i)T^{2} \) |
| 59 | \( 1 + (-4.02e3 - 1.23e4i)T + (-5.78e8 + 4.20e8i)T^{2} \) |
| 61 | \( 1 + (-6.66e3 + 2.05e4i)T + (-6.83e8 - 4.96e8i)T^{2} \) |
| 67 | \( 1 + (-2.74e4 + 1.99e4i)T + (4.17e8 - 1.28e9i)T^{2} \) |
| 71 | \( 1 + (-2.83e4 - 2.06e4i)T + (5.57e8 + 1.71e9i)T^{2} \) |
| 73 | \( 1 + (-4.38e3 + 1.34e4i)T + (-1.67e9 - 1.21e9i)T^{2} \) |
| 79 | \( 1 + (7.80e3 + 5.66e3i)T + (9.50e8 + 2.92e9i)T^{2} \) |
| 83 | \( 1 + (-8.73e4 + 6.34e4i)T + (1.21e9 - 3.74e9i)T^{2} \) |
| 89 | \( 1 + (952. - 2.93e3i)T + (-4.51e9 - 3.28e9i)T^{2} \) |
| 97 | \( 1 + (1.25e5 + 9.08e4i)T + (2.65e9 + 8.16e9i)T^{2} \) |
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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
−16.61041072756371500176719935271, −15.14453217183621086358615322210, −13.59816223614398581247122830232, −12.36410104251372619709963157713, −11.46164138548783377422733240817, −9.756388619543797690343864528684, −8.257466909260569228204956635748, −6.28600953013728093919096365221, −4.08872421198175280303668403692, −1.76698137145928911475838452587,
2.34395118272543967127051689619, 5.33277099435572635140069166143, 6.70193746607678006892737387876, 8.175369909975751364034127098932, 10.49394643005477934211230785170, 11.06380151373291494564875227158, 13.26755338293167394918566183791, 14.43839741723446617301403449069, 15.25490219570178385243657934996, 16.60754367339015226889103482325
