| L(s) = 1 | + (0.235 − 0.151i)2-s + (0.0812 − 0.565i)3-s + (−1.62 + 3.56i)4-s + (0.629 + 2.14i)5-s + (−0.0664 − 0.145i)6-s + (−3.04 + 2.63i)7-s + (0.315 + 2.19i)8-s + (8.32 + 2.44i)9-s + (0.473 + 0.410i)10-s + (−9.09 + 14.1i)11-s + (1.88 + 1.21i)12-s + (−0.630 + 0.727i)13-s + (−0.318 + 1.08i)14-s + (1.26 − 0.181i)15-s + (−9.86 − 11.3i)16-s + (20.9 − 9.55i)17-s + ⋯ |
| L(s) = 1 | + (0.117 − 0.0757i)2-s + (0.0270 − 0.188i)3-s + (−0.407 + 0.891i)4-s + (0.125 + 0.429i)5-s + (−0.0110 − 0.0242i)6-s + (−0.435 + 0.376i)7-s + (0.0394 + 0.274i)8-s + (0.924 + 0.271i)9-s + (0.0473 + 0.0410i)10-s + (−0.826 + 1.28i)11-s + (0.156 + 0.100i)12-s + (−0.0484 + 0.0559i)13-s + (−0.0227 + 0.0773i)14-s + (0.0842 − 0.0121i)15-s + (−0.616 − 0.711i)16-s + (1.23 − 0.561i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.204 - 0.978i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.204 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.985925 + 0.801297i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.985925 + 0.801297i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-0.629 - 2.14i)T \) |
| 23 | \( 1 + (22.9 - 0.326i)T \) |
| good | 2 | \( 1 + (-0.235 + 0.151i)T + (1.66 - 3.63i)T^{2} \) |
| 3 | \( 1 + (-0.0812 + 0.565i)T + (-8.63 - 2.53i)T^{2} \) |
| 7 | \( 1 + (3.04 - 2.63i)T + (6.97 - 48.5i)T^{2} \) |
| 11 | \( 1 + (9.09 - 14.1i)T + (-50.2 - 110. i)T^{2} \) |
| 13 | \( 1 + (0.630 - 0.727i)T + (-24.0 - 167. i)T^{2} \) |
| 17 | \( 1 + (-20.9 + 9.55i)T + (189. - 218. i)T^{2} \) |
| 19 | \( 1 + (-23.8 - 10.8i)T + (236. + 272. i)T^{2} \) |
| 29 | \( 1 + (14.6 + 32.0i)T + (-550. + 635. i)T^{2} \) |
| 31 | \( 1 + (-0.766 - 5.32i)T + (-922. + 270. i)T^{2} \) |
| 37 | \( 1 + (-13.4 + 45.7i)T + (-1.15e3 - 740. i)T^{2} \) |
| 41 | \( 1 + (17.5 - 5.16i)T + (1.41e3 - 908. i)T^{2} \) |
| 43 | \( 1 + (-30.5 - 4.39i)T + (1.77e3 + 520. i)T^{2} \) |
| 47 | \( 1 - 66.9T + 2.20e3T^{2} \) |
| 53 | \( 1 + (-70.0 + 60.7i)T + (399. - 2.78e3i)T^{2} \) |
| 59 | \( 1 + (-6.96 + 8.03i)T + (-495. - 3.44e3i)T^{2} \) |
| 61 | \( 1 + (77.2 - 11.1i)T + (3.57e3 - 1.04e3i)T^{2} \) |
| 67 | \( 1 + (-60.0 - 93.3i)T + (-1.86e3 + 4.08e3i)T^{2} \) |
| 71 | \( 1 + (26.9 - 17.3i)T + (2.09e3 - 4.58e3i)T^{2} \) |
| 73 | \( 1 + (34.9 - 76.6i)T + (-3.48e3 - 4.02e3i)T^{2} \) |
| 79 | \( 1 + (-49.2 - 42.7i)T + (888. + 6.17e3i)T^{2} \) |
| 83 | \( 1 + (-29.5 + 100. i)T + (-5.79e3 - 3.72e3i)T^{2} \) |
| 89 | \( 1 + (-67.0 - 9.64i)T + (7.60e3 + 2.23e3i)T^{2} \) |
| 97 | \( 1 + (-5.13 - 17.5i)T + (-7.91e3 + 5.08e3i)T^{2} \) |
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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
−13.39307342243002490973968941426, −12.50130700317833067494777319162, −11.88673300791527621884076544792, −10.14679952201894666920887907730, −9.523263669532708286421705459579, −7.74759746076293230472508073041, −7.33238676841013757990763723633, −5.47365329031593242796061126224, −4.00682392726572735667950092563, −2.47529808746195764685526775600,
0.964889741924944756231520281050, 3.60274936169892771187652003973, 5.10668357435808962843057817907, 6.07007642382251584287835924151, 7.60925493948891053677435798174, 9.029612325615653457065821497459, 10.00015962876889420927576520062, 10.67431655704971593178309641232, 12.23659340539756979643199765591, 13.38508435509932326220360663382
