| L(s) = 1 | + (−1.05 + 3.23i)5-s + (−0.703 + 0.510i)7-s + (−3.16 + 0.998i)11-s + (0.866 + 2.66i)13-s + (2.25 − 6.93i)17-s + (−1.92 − 1.40i)19-s − 7.44·23-s + (−5.34 − 3.88i)25-s + (−5.93 + 4.31i)29-s + (−0.816 − 2.51i)31-s + (−0.914 − 2.81i)35-s + (−0.834 + 0.605i)37-s + (2.34 + 1.70i)41-s + 2.18·43-s + (−3.19 − 2.31i)47-s + ⋯ |
| L(s) = 1 | + (−0.470 + 1.44i)5-s + (−0.265 + 0.193i)7-s + (−0.953 + 0.301i)11-s + (0.240 + 0.739i)13-s + (0.546 − 1.68i)17-s + (−0.442 − 0.321i)19-s − 1.55·23-s + (−1.06 − 0.776i)25-s + (−1.10 + 0.801i)29-s + (−0.146 − 0.451i)31-s + (−0.154 − 0.475i)35-s + (−0.137 + 0.0996i)37-s + (0.365 + 0.265i)41-s + 0.333·43-s + (−0.465 − 0.338i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0540i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 - 0.0540i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0137548 + 0.508493i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0137548 + 0.508493i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 + (3.16 - 0.998i)T \) |
| good | 5 | \( 1 + (1.05 - 3.23i)T + (-4.04 - 2.93i)T^{2} \) |
| 7 | \( 1 + (0.703 - 0.510i)T + (2.16 - 6.65i)T^{2} \) |
| 13 | \( 1 + (-0.866 - 2.66i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (-2.25 + 6.93i)T + (-13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (1.92 + 1.40i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + 7.44T + 23T^{2} \) |
| 29 | \( 1 + (5.93 - 4.31i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (0.816 + 2.51i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (0.834 - 0.605i)T + (11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (-2.34 - 1.70i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 - 2.18T + 43T^{2} \) |
| 47 | \( 1 + (3.19 + 2.31i)T + (14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (-2.19 - 6.76i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (-6.54 + 4.75i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (0.832 - 2.56i)T + (-49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 + 13.7T + 67T^{2} \) |
| 71 | \( 1 + (-0.596 + 1.83i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (4.03 - 2.93i)T + (22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (-4.15 - 12.7i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (-0.0601 + 0.185i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + 4.18T + 89T^{2} \) |
| 97 | \( 1 + (-1.54 - 4.74i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−10.72593334246056543349864218649, −9.938856178707625906927653122925, −9.150888610911907698887840174147, −7.83739660117399585697164107296, −7.30660267415812276049613090515, −6.49456839048564368049388303584, −5.47910652794093220208913384587, −4.21940943522589201403484630174, −3.13016710645240185708633184290, −2.29372648021937097200733043607,
0.24118586927051563360971080410, 1.78951835528885506111808824122, 3.53548500413385691156175991579, 4.30763373822758850330302241768, 5.49956188151461459044100325038, 6.03182106436483903391961855132, 7.67496929403051075164763646838, 8.158874862472973242037122841804, 8.784331246963031317258875309273, 9.992523948752305038816289419361
