| L(s) = 1 | + (−1.39 − 0.208i)2-s + (0.0768 + 1.73i)3-s + (1.91 + 0.582i)4-s + (−2.08 + 2.38i)5-s + (0.252 − 2.43i)6-s + (2.53 − 0.333i)7-s + (−2.55 − 1.21i)8-s + (−2.98 + 0.266i)9-s + (3.41 − 2.89i)10-s + (−0.249 + 0.506i)11-s + (−0.860 + 3.35i)12-s + (−1.58 + 4.67i)13-s + (−3.61 − 0.0608i)14-s + (−4.27 − 3.42i)15-s + (3.32 + 2.22i)16-s + (1.15 − 1.15i)17-s + ⋯ |
| L(s) = 1 | + (−0.989 − 0.147i)2-s + (0.0443 + 0.999i)3-s + (0.956 + 0.291i)4-s + (−0.933 + 1.06i)5-s + (0.103 − 0.994i)6-s + (0.958 − 0.126i)7-s + (−0.903 − 0.428i)8-s + (−0.996 + 0.0886i)9-s + (1.08 − 0.915i)10-s + (−0.0752 + 0.152i)11-s + (−0.248 + 0.968i)12-s + (−0.439 + 1.29i)13-s + (−0.966 − 0.0162i)14-s + (−1.10 − 0.885i)15-s + (0.830 + 0.557i)16-s + (0.280 − 0.280i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.00130i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.00130i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.000361988 + 0.555352i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.000361988 + 0.555352i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.39 + 0.208i)T \) |
| 3 | \( 1 + (-0.0768 - 1.73i)T \) |
| good | 5 | \( 1 + (2.08 - 2.38i)T + (-0.652 - 4.95i)T^{2} \) |
| 7 | \( 1 + (-2.53 + 0.333i)T + (6.76 - 1.81i)T^{2} \) |
| 11 | \( 1 + (0.249 - 0.506i)T + (-6.69 - 8.72i)T^{2} \) |
| 13 | \( 1 + (1.58 - 4.67i)T + (-10.3 - 7.91i)T^{2} \) |
| 17 | \( 1 + (-1.15 + 1.15i)T - 17iT^{2} \) |
| 19 | \( 1 + (-0.747 - 3.75i)T + (-17.5 + 7.27i)T^{2} \) |
| 23 | \( 1 + (-0.00142 + 0.0108i)T + (-22.2 - 5.95i)T^{2} \) |
| 29 | \( 1 + (4.49 + 0.294i)T + (28.7 + 3.78i)T^{2} \) |
| 31 | \( 1 + (3.52 + 2.03i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.974 + 4.90i)T + (-34.1 - 14.1i)T^{2} \) |
| 41 | \( 1 + (-0.378 + 2.87i)T + (-39.6 - 10.6i)T^{2} \) |
| 43 | \( 1 + (6.76 + 3.33i)T + (26.1 + 34.1i)T^{2} \) |
| 47 | \( 1 + (-3.55 - 0.952i)T + (40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (-7.04 - 10.5i)T + (-20.2 + 48.9i)T^{2} \) |
| 59 | \( 1 + (-0.853 + 0.973i)T + (-7.70 - 58.4i)T^{2} \) |
| 61 | \( 1 + (-0.584 + 8.91i)T + (-60.4 - 7.96i)T^{2} \) |
| 67 | \( 1 + (7.01 - 3.45i)T + (40.7 - 53.1i)T^{2} \) |
| 71 | \( 1 + (4.98 - 12.0i)T + (-50.2 - 50.2i)T^{2} \) |
| 73 | \( 1 + (6.20 + 14.9i)T + (-51.6 + 51.6i)T^{2} \) |
| 79 | \( 1 + (4.48 - 16.7i)T + (-68.4 - 39.5i)T^{2} \) |
| 83 | \( 1 + (4.91 - 4.30i)T + (10.8 - 82.2i)T^{2} \) |
| 89 | \( 1 + (-3.90 - 1.61i)T + (62.9 + 62.9i)T^{2} \) |
| 97 | \( 1 + (-12.9 + 7.50i)T + (48.5 - 84.0i)T^{2} \) |
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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
−11.09421316169065968935331606388, −10.33481457165507448245192403387, −9.495357623243124323771705427442, −8.613604544962218415424213235603, −7.64651590657266574927925957086, −7.12015819851608117232372384569, −5.74738765279955318943929587674, −4.30074763864824357260687066005, −3.44811833607096989960938479718, −2.13533742173739404829178110945,
0.42986154507983399988423184352, 1.59578944309933010249539563646, 3.08346885263933403435949532251, 4.94090908035187863431183839672, 5.75333290547199750682515108997, 7.16351357716178053734515592637, 7.80858821877946929564210485848, 8.371474390403219195799918529938, 8.962483156394720355255285862129, 10.29129858129538001228730157977
