| L(s) = 1 | + (−1.48 − 4.58i)2-s + (−12.3 + 8.93i)4-s + (−5.89 − 9.49i)5-s + 1.13·7-s + (28.0 + 20.4i)8-s + (−34.7 + 41.1i)10-s + (17.7 + 54.7i)11-s + (−12.9 + 39.9i)13-s + (−1.68 − 5.19i)14-s + (14.0 − 43.3i)16-s + (−87.1 − 63.3i)17-s + (43.1 + 31.3i)19-s + (157. + 64.1i)20-s + (224. − 162. i)22-s + (5.32 + 16.3i)23-s + ⋯ |
| L(s) = 1 | + (−0.526 − 1.61i)2-s + (−1.53 + 1.11i)4-s + (−0.527 − 0.849i)5-s + 0.0612·7-s + (1.24 + 0.901i)8-s + (−1.09 + 1.30i)10-s + (0.487 + 1.49i)11-s + (−0.276 + 0.851i)13-s + (−0.0322 − 0.0992i)14-s + (0.219 − 0.677i)16-s + (−1.24 − 0.903i)17-s + (0.520 + 0.378i)19-s + (1.76 + 0.716i)20-s + (2.17 − 1.57i)22-s + (0.0482 + 0.148i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.922 + 0.386i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.922 + 0.386i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.666689 - 0.133894i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.666689 - 0.133894i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (5.89 + 9.49i)T \) |
| good | 2 | \( 1 + (1.48 + 4.58i)T + (-6.47 + 4.70i)T^{2} \) |
| 7 | \( 1 - 1.13T + 343T^{2} \) |
| 11 | \( 1 + (-17.7 - 54.7i)T + (-1.07e3 + 782. i)T^{2} \) |
| 13 | \( 1 + (12.9 - 39.9i)T + (-1.77e3 - 1.29e3i)T^{2} \) |
| 17 | \( 1 + (87.1 + 63.3i)T + (1.51e3 + 4.67e3i)T^{2} \) |
| 19 | \( 1 + (-43.1 - 31.3i)T + (2.11e3 + 6.52e3i)T^{2} \) |
| 23 | \( 1 + (-5.32 - 16.3i)T + (-9.84e3 + 7.15e3i)T^{2} \) |
| 29 | \( 1 + (-113. + 82.4i)T + (7.53e3 - 2.31e4i)T^{2} \) |
| 31 | \( 1 + (-196. - 142. i)T + (9.20e3 + 2.83e4i)T^{2} \) |
| 37 | \( 1 + (-107. + 332. i)T + (-4.09e4 - 2.97e4i)T^{2} \) |
| 41 | \( 1 + (-2.64 + 8.13i)T + (-5.57e4 - 4.05e4i)T^{2} \) |
| 43 | \( 1 - 111.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (261. - 189. i)T + (3.20e4 - 9.87e4i)T^{2} \) |
| 53 | \( 1 + (332. - 241. i)T + (4.60e4 - 1.41e5i)T^{2} \) |
| 59 | \( 1 + (205. - 633. i)T + (-1.66e5 - 1.20e5i)T^{2} \) |
| 61 | \( 1 + (-91.5 - 281. i)T + (-1.83e5 + 1.33e5i)T^{2} \) |
| 67 | \( 1 + (-679. - 494. i)T + (9.29e4 + 2.86e5i)T^{2} \) |
| 71 | \( 1 + (45.5 - 33.1i)T + (1.10e5 - 3.40e5i)T^{2} \) |
| 73 | \( 1 + (-66.9 - 206. i)T + (-3.14e5 + 2.28e5i)T^{2} \) |
| 79 | \( 1 + (339. - 246. i)T + (1.52e5 - 4.68e5i)T^{2} \) |
| 83 | \( 1 + (-1.17e3 - 856. i)T + (1.76e5 + 5.43e5i)T^{2} \) |
| 89 | \( 1 + (-378. - 1.16e3i)T + (-5.70e5 + 4.14e5i)T^{2} \) |
| 97 | \( 1 + (419. - 305. i)T + (2.82e5 - 8.68e5i)T^{2} \) |
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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
−11.87030617131974596872275069171, −10.93345513481539036699134402556, −9.575722785332930872699988840107, −9.336534976092893209630051862813, −8.166575186555890847368574766455, −6.93174740840810312600034868720, −4.72333759710539541638788199316, −4.13651079308776632357913018272, −2.44163088480707814780508163815, −1.21479504714986733763069362344,
0.38741010214262435573950824241, 3.20017785781522538716518202529, 4.79089338421846586230451805775, 6.23150930084054400965695258765, 6.62116416010225263500395972594, 8.018553339102727446340160195596, 8.364351477928532013276965886548, 9.629839467740796386336766275815, 10.76557512487320113835511548184, 11.62240530506129594823698712057
