| L(s) = 1 | + (−2.23 + 3.07i)2-s + (8.89 − 2.89i)3-s + (−2.00 − 6.16i)4-s + (−3.48 + 10.6i)5-s + (−11.0 + 33.8i)6-s − 4.54i·7-s + (−5.47 − 1.77i)8-s + (48.9 − 35.5i)9-s + (−24.9 − 34.4i)10-s + (10.9 + 7.94i)11-s + (−35.6 − 49.1i)12-s + (−40.2 − 55.3i)13-s + (14.0 + 10.1i)14-s + (−0.265 + 104. i)15-s + (59.7 − 43.3i)16-s + (−35.1 − 11.4i)17-s + ⋯ |
| L(s) = 1 | + (−0.790 + 1.08i)2-s + (1.71 − 0.556i)3-s + (−0.250 − 0.771i)4-s + (−0.311 + 0.950i)5-s + (−0.748 + 2.30i)6-s − 0.245i·7-s + (−0.241 − 0.0786i)8-s + (1.81 − 1.31i)9-s + (−0.788 − 1.09i)10-s + (0.299 + 0.217i)11-s + (−0.858 − 1.18i)12-s + (−0.858 − 1.18i)13-s + (0.267 + 0.194i)14-s + (−0.00456 + 1.80i)15-s + (0.933 − 0.677i)16-s + (−0.501 − 0.163i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.430 - 0.902i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.430 - 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.995761 + 0.628392i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.995761 + 0.628392i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (3.48 - 10.6i)T \) |
| good | 2 | \( 1 + (2.23 - 3.07i)T + (-2.47 - 7.60i)T^{2} \) |
| 3 | \( 1 + (-8.89 + 2.89i)T + (21.8 - 15.8i)T^{2} \) |
| 7 | \( 1 + 4.54iT - 343T^{2} \) |
| 11 | \( 1 + (-10.9 - 7.94i)T + (411. + 1.26e3i)T^{2} \) |
| 13 | \( 1 + (40.2 + 55.3i)T + (-678. + 2.08e3i)T^{2} \) |
| 17 | \( 1 + (35.1 + 11.4i)T + (3.97e3 + 2.88e3i)T^{2} \) |
| 19 | \( 1 + (-0.479 + 1.47i)T + (-5.54e3 - 4.03e3i)T^{2} \) |
| 23 | \( 1 + (-0.969 + 1.33i)T + (-3.75e3 - 1.15e4i)T^{2} \) |
| 29 | \( 1 + (-49.0 - 151. i)T + (-1.97e4 + 1.43e4i)T^{2} \) |
| 31 | \( 1 + (83.6 - 257. i)T + (-2.41e4 - 1.75e4i)T^{2} \) |
| 37 | \( 1 + (-81.9 - 112. i)T + (-1.56e4 + 4.81e4i)T^{2} \) |
| 41 | \( 1 + (-80.5 + 58.5i)T + (2.12e4 - 6.55e4i)T^{2} \) |
| 43 | \( 1 + 135. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + (171. - 55.5i)T + (8.39e4 - 6.10e4i)T^{2} \) |
| 53 | \( 1 + (303. - 98.5i)T + (1.20e5 - 8.75e4i)T^{2} \) |
| 59 | \( 1 + (119. - 86.4i)T + (6.34e4 - 1.95e5i)T^{2} \) |
| 61 | \( 1 + (-180. - 131. i)T + (7.01e4 + 2.15e5i)T^{2} \) |
| 67 | \( 1 + (-842. - 273. i)T + (2.43e5 + 1.76e5i)T^{2} \) |
| 71 | \( 1 + (163. + 503. i)T + (-2.89e5 + 2.10e5i)T^{2} \) |
| 73 | \( 1 + (210. - 289. i)T + (-1.20e5 - 3.69e5i)T^{2} \) |
| 79 | \( 1 + (243. + 750. i)T + (-3.98e5 + 2.89e5i)T^{2} \) |
| 83 | \( 1 + (855. + 277. i)T + (4.62e5 + 3.36e5i)T^{2} \) |
| 89 | \( 1 + (-68.1 - 49.4i)T + (2.17e5 + 6.70e5i)T^{2} \) |
| 97 | \( 1 + (-851. + 276. i)T + (7.38e5 - 5.36e5i)T^{2} \) |
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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
−17.60238324761301021043307853169, −15.74442775747938334098711626425, −14.89781640456995538364223914745, −14.17916274583340393822545320171, −12.57018464124040685138226802964, −10.09853915275566794157064958309, −8.756549415279338481648401331442, −7.62123157486148577402218502506, −6.89968040184411503218847977811, −3.07662505604683658042321496622,
2.19502570510917796133452587709, 4.12174438086467098700793093211, 8.054730186583226078695297165860, 9.130632484041134975084128511007, 9.686730154729834633008953261009, 11.52955857274077150514068726300, 12.95323561883219326226108704963, 14.39653720128968829670795483597, 15.55095502826326490681479830469, 16.94704846828321445784833729683
