| L(s) = 1 | + (1.18 + 1.26i)3-s + 5-s + (2 + 1.73i)7-s + (−0.186 + 2.99i)9-s + 2.52i·11-s + 4.10i·13-s + (1.18 + 1.26i)15-s + 4.37·17-s − 3.46i·19-s + (0.186 + 4.57i)21-s − 8.51i·23-s + 25-s + (−4.00 + 3.31i)27-s + 0.939i·29-s + 3.46i·31-s + ⋯ |
| L(s) = 1 | + (0.684 + 0.728i)3-s + 0.447·5-s + (0.755 + 0.654i)7-s + (−0.0620 + 0.998i)9-s + 0.761i·11-s + 1.13i·13-s + (0.306 + 0.325i)15-s + 1.06·17-s − 0.794i·19-s + (0.0406 + 0.999i)21-s − 1.77i·23-s + 0.200·25-s + (−0.769 + 0.638i)27-s + 0.174i·29-s + 0.622i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0406 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0406 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.575328227\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.575328227\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.18 - 1.26i)T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + (-2 - 1.73i)T \) |
| good | 11 | \( 1 - 2.52iT - 11T^{2} \) |
| 13 | \( 1 - 4.10iT - 13T^{2} \) |
| 17 | \( 1 - 4.37T + 17T^{2} \) |
| 19 | \( 1 + 3.46iT - 19T^{2} \) |
| 23 | \( 1 + 8.51iT - 23T^{2} \) |
| 29 | \( 1 - 0.939iT - 29T^{2} \) |
| 31 | \( 1 - 3.46iT - 31T^{2} \) |
| 37 | \( 1 + 6.74T + 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 4.74T + 43T^{2} \) |
| 47 | \( 1 - 1.62T + 47T^{2} \) |
| 53 | \( 1 + 1.87iT - 53T^{2} \) |
| 59 | \( 1 - 8.74T + 59T^{2} \) |
| 61 | \( 1 + 6.92iT - 61T^{2} \) |
| 67 | \( 1 + 4.74T + 67T^{2} \) |
| 71 | \( 1 + 0.294iT - 71T^{2} \) |
| 73 | \( 1 + 6.92iT - 73T^{2} \) |
| 79 | \( 1 - 2.37T + 79T^{2} \) |
| 83 | \( 1 - 17.4T + 83T^{2} \) |
| 89 | \( 1 - 14.7T + 89T^{2} \) |
| 97 | \( 1 - 11.0iT - 97T^{2} \) |
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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
−9.438242648325353165482707984201, −8.822575378031471484475864479674, −8.251105632794963590368504380644, −7.23055321861020345182662547133, −6.38009393125032739416427796896, −4.97434084130805895185228047897, −4.89054616005883675093584284413, −3.63943847877238736800803743141, −2.48347355948800679712497443321, −1.76258207761313104480763683610,
0.949243845249086983080356334854, 1.82793391029484546896910992866, 3.19302991078330830961033496169, 3.74834876046584214026066090217, 5.31345107649271014879651025619, 5.81164826170481572392949368662, 6.94008390048820087339990649830, 7.81067295396792885960194970919, 8.076724171709400873716758310574, 9.019158213140981548216300047336
