| L(s) = 1 | + (−0.639 − 0.737i)2-s + (−5.88 − 3.78i)3-s + (1.00 − 6.97i)4-s + (0.787 − 1.72i)5-s + (0.971 + 6.75i)6-s + (10.9 + 3.22i)7-s + (−12.3 + 7.94i)8-s + (9.11 + 19.9i)9-s + (−1.77 + 0.521i)10-s + (15.8 − 18.3i)11-s + (−32.2 + 37.2i)12-s + (49.2 − 14.4i)13-s + (−4.64 − 10.1i)14-s + (−11.1 + 7.17i)15-s + (−40.3 − 11.8i)16-s + (11.8 + 82.6i)17-s + ⋯ |
| L(s) = 1 | + (−0.225 − 0.260i)2-s + (−1.13 − 0.727i)3-s + (0.125 − 0.871i)4-s + (0.0704 − 0.154i)5-s + (0.0661 + 0.459i)6-s + (0.592 + 0.174i)7-s + (−0.546 + 0.350i)8-s + (0.337 + 0.739i)9-s + (−0.0561 + 0.0164i)10-s + (0.435 − 0.502i)11-s + (−0.776 + 0.896i)12-s + (1.04 − 0.308i)13-s + (−0.0885 − 0.193i)14-s + (−0.192 + 0.123i)15-s + (−0.630 − 0.185i)16-s + (0.169 + 1.17i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.338 + 0.941i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.338 + 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.442068 - 0.628566i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.442068 - 0.628566i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 + (103. + 38.5i)T \) |
| good | 2 | \( 1 + (0.639 + 0.737i)T + (-1.13 + 7.91i)T^{2} \) |
| 3 | \( 1 + (5.88 + 3.78i)T + (11.2 + 24.5i)T^{2} \) |
| 5 | \( 1 + (-0.787 + 1.72i)T + (-81.8 - 94.4i)T^{2} \) |
| 7 | \( 1 + (-10.9 - 3.22i)T + (288. + 185. i)T^{2} \) |
| 11 | \( 1 + (-15.8 + 18.3i)T + (-189. - 1.31e3i)T^{2} \) |
| 13 | \( 1 + (-49.2 + 14.4i)T + (1.84e3 - 1.18e3i)T^{2} \) |
| 17 | \( 1 + (-11.8 - 82.6i)T + (-4.71e3 + 1.38e3i)T^{2} \) |
| 19 | \( 1 + (-21.4 + 149. i)T + (-6.58e3 - 1.93e3i)T^{2} \) |
| 29 | \( 1 + (-35.8 - 249. i)T + (-2.34e4 + 6.87e3i)T^{2} \) |
| 31 | \( 1 + (-36.8 + 23.6i)T + (1.23e4 - 2.70e4i)T^{2} \) |
| 37 | \( 1 + (-14.8 - 32.5i)T + (-3.31e4 + 3.82e4i)T^{2} \) |
| 41 | \( 1 + (-138. + 303. i)T + (-4.51e4 - 5.20e4i)T^{2} \) |
| 43 | \( 1 + (-247. - 158. i)T + (3.30e4 + 7.23e4i)T^{2} \) |
| 47 | \( 1 + 103.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (203. + 59.6i)T + (1.25e5 + 8.04e4i)T^{2} \) |
| 59 | \( 1 + (-110. + 32.3i)T + (1.72e5 - 1.11e5i)T^{2} \) |
| 61 | \( 1 + (-154. + 99.5i)T + (9.42e4 - 2.06e5i)T^{2} \) |
| 67 | \( 1 + (-357. - 412. i)T + (-4.28e4 + 2.97e5i)T^{2} \) |
| 71 | \( 1 + (-381. - 440. i)T + (-5.09e4 + 3.54e5i)T^{2} \) |
| 73 | \( 1 + (6.74 - 46.9i)T + (-3.73e5 - 1.09e5i)T^{2} \) |
| 79 | \( 1 + (1.19e3 - 352. i)T + (4.14e5 - 2.66e5i)T^{2} \) |
| 83 | \( 1 + (315. + 691. i)T + (-3.74e5 + 4.32e5i)T^{2} \) |
| 89 | \( 1 + (-661. - 424. i)T + (2.92e5 + 6.41e5i)T^{2} \) |
| 97 | \( 1 + (232. - 509. i)T + (-5.97e5 - 6.89e5i)T^{2} \) |
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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
−17.41259261691480845073341462156, −15.89154138400024159065054361706, −14.42902754473906355755178780908, −12.89471988747001164395623572483, −11.43670758653091740299283914735, −10.79224980314215150481762256457, −8.790833094673480631465999010178, −6.56659498545760676370792492346, −5.41690941727131998945448111280, −1.15468021383333519785073374713,
4.21576610073768327004048051488, 6.17624657730425106531700673688, 7.961601967673935168217832708761, 9.815804520583073384085820132759, 11.32662722801887366722112639638, 12.14082634428788323755422924698, 14.07200689758923659779575546727, 15.84100080475962037737304364404, 16.50174756096093863639352970753, 17.54709209388951187688416033781
