| L(s) = 1 | + (3.53 + 1.61i)2-s + (0.969 − 3.30i)3-s + (7.26 + 8.38i)4-s + (−4.90 − 0.959i)5-s + (8.75 − 10.1i)6-s + (0.693 − 4.82i)7-s + (7.77 + 26.4i)8-s + (−2.38 − 1.53i)9-s + (−15.7 − 11.3i)10-s + (−10.6 + 4.85i)11-s + (34.7 − 15.8i)12-s + (−10.6 + 1.52i)13-s + (10.2 − 15.9i)14-s + (−7.92 + 15.2i)15-s + (−8.93 + 62.1i)16-s + (11.7 − 13.5i)17-s + ⋯ |
| L(s) = 1 | + (1.76 + 0.807i)2-s + (0.323 − 1.10i)3-s + (1.81 + 2.09i)4-s + (−0.981 − 0.191i)5-s + (1.45 − 1.68i)6-s + (0.0991 − 0.689i)7-s + (0.971 + 3.30i)8-s + (−0.264 − 0.170i)9-s + (−1.57 − 1.13i)10-s + (−0.965 + 0.441i)11-s + (2.89 − 1.32i)12-s + (−0.816 + 0.117i)13-s + (0.731 − 1.13i)14-s + (−0.528 + 1.01i)15-s + (−0.558 + 3.88i)16-s + (0.691 − 0.797i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.924 - 0.380i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.924 - 0.380i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.21716 + 0.636645i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.21716 + 0.636645i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (4.90 + 0.959i)T \) |
| 23 | \( 1 + (-5.90 + 22.2i)T \) |
| good | 2 | \( 1 + (-3.53 - 1.61i)T + (2.61 + 3.02i)T^{2} \) |
| 3 | \( 1 + (-0.969 + 3.30i)T + (-7.57 - 4.86i)T^{2} \) |
| 7 | \( 1 + (-0.693 + 4.82i)T + (-47.0 - 13.8i)T^{2} \) |
| 11 | \( 1 + (10.6 - 4.85i)T + (79.2 - 91.4i)T^{2} \) |
| 13 | \( 1 + (10.6 - 1.52i)T + (162. - 47.6i)T^{2} \) |
| 17 | \( 1 + (-11.7 + 13.5i)T + (-41.1 - 286. i)T^{2} \) |
| 19 | \( 1 + (9.78 - 8.48i)T + (51.3 - 357. i)T^{2} \) |
| 29 | \( 1 + (11.7 - 13.5i)T + (-119. - 832. i)T^{2} \) |
| 31 | \( 1 + (34.9 - 10.2i)T + (808. - 519. i)T^{2} \) |
| 37 | \( 1 + (-30.9 - 19.8i)T + (568. + 1.24e3i)T^{2} \) |
| 41 | \( 1 + (-58.9 + 37.8i)T + (698. - 1.52e3i)T^{2} \) |
| 43 | \( 1 + (13.7 + 4.04i)T + (1.55e3 + 9.99e2i)T^{2} \) |
| 47 | \( 1 + 23.8iT - 2.20e3T^{2} \) |
| 53 | \( 1 + (-6.74 + 46.9i)T + (-2.69e3 - 791. i)T^{2} \) |
| 59 | \( 1 + (-1.19 - 8.29i)T + (-3.33e3 + 980. i)T^{2} \) |
| 61 | \( 1 + (-12.2 - 41.6i)T + (-3.13e3 + 2.01e3i)T^{2} \) |
| 67 | \( 1 + (21.4 - 46.8i)T + (-2.93e3 - 3.39e3i)T^{2} \) |
| 71 | \( 1 + (-4.48 + 9.82i)T + (-3.30e3 - 3.80e3i)T^{2} \) |
| 73 | \( 1 + (38.8 - 33.6i)T + (758. - 5.27e3i)T^{2} \) |
| 79 | \( 1 + (-51.6 + 7.43i)T + (5.98e3 - 1.75e3i)T^{2} \) |
| 83 | \( 1 + (-61.3 - 39.4i)T + (2.86e3 + 6.26e3i)T^{2} \) |
| 89 | \( 1 + (41.1 - 140. i)T + (-6.66e3 - 4.28e3i)T^{2} \) |
| 97 | \( 1 + (69.9 - 44.9i)T + (3.90e3 - 8.55e3i)T^{2} \) |
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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
−13.33344774118823542940591198912, −12.60850545313648173035215653149, −12.10613914925044075259522220197, −10.79653617048090069658199281925, −8.176406351938824764466013071873, −7.40639971053859358003335626958, −6.98388444094811982052120906299, −5.23649932934039483959570667133, −4.14755528416603966668659757605, −2.61089136601556586017774900366,
2.71701486672729651614690935454, 3.74226803572482167597622999217, 4.76034435627250992187768689518, 5.80526521011861904246143884189, 7.59849327741172251442620906623, 9.465122072523248934539028777744, 10.61916324266827523274486606184, 11.24703198577738439419433372056, 12.35267759747443440250375663200, 13.07045159576339780580940740714
