| L(s) = 1 | + (−3.24 + 2.08i)2-s + (1.21 − 8.45i)3-s + (2.84 − 6.23i)4-s + (12.7 − 3.75i)5-s + (13.6 + 29.9i)6-s + (−11.3 − 13.0i)7-s + (−0.631 − 4.38i)8-s + (−44.1 − 12.9i)9-s + (−33.6 + 38.8i)10-s + (45.2 + 29.0i)11-s + (−49.2 − 31.6i)12-s + (−27.2 + 31.4i)13-s + (64.0 + 18.8i)14-s + (−16.2 − 112. i)15-s + (47.0 + 54.3i)16-s + (14.8 + 32.4i)17-s + ⋯ |
| L(s) = 1 | + (−1.14 + 0.736i)2-s + (0.234 − 1.62i)3-s + (0.355 − 0.778i)4-s + (1.14 − 0.336i)5-s + (0.930 + 2.03i)6-s + (−0.612 − 0.707i)7-s + (−0.0278 − 0.193i)8-s + (−1.63 − 0.480i)9-s + (−1.06 + 1.22i)10-s + (1.24 + 0.796i)11-s + (−1.18 − 0.761i)12-s + (−0.581 + 0.671i)13-s + (1.22 + 0.359i)14-s + (−0.279 − 1.94i)15-s + (0.735 + 0.848i)16-s + (0.211 + 0.463i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.661 + 0.749i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.661 + 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.697359 - 0.314700i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.697359 - 0.314700i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 + (-89.7 + 64.1i)T \) |
| good | 2 | \( 1 + (3.24 - 2.08i)T + (3.32 - 7.27i)T^{2} \) |
| 3 | \( 1 + (-1.21 + 8.45i)T + (-25.9 - 7.60i)T^{2} \) |
| 5 | \( 1 + (-12.7 + 3.75i)T + (105. - 67.5i)T^{2} \) |
| 7 | \( 1 + (11.3 + 13.0i)T + (-48.8 + 339. i)T^{2} \) |
| 11 | \( 1 + (-45.2 - 29.0i)T + (552. + 1.21e3i)T^{2} \) |
| 13 | \( 1 + (27.2 - 31.4i)T + (-312. - 2.17e3i)T^{2} \) |
| 17 | \( 1 + (-14.8 - 32.4i)T + (-3.21e3 + 3.71e3i)T^{2} \) |
| 19 | \( 1 + (-8.29 + 18.1i)T + (-4.49e3 - 5.18e3i)T^{2} \) |
| 29 | \( 1 + (-77.8 - 170. i)T + (-1.59e4 + 1.84e4i)T^{2} \) |
| 31 | \( 1 + (0.302 + 2.10i)T + (-2.85e4 + 8.39e3i)T^{2} \) |
| 37 | \( 1 + (-99.0 - 29.0i)T + (4.26e4 + 2.73e4i)T^{2} \) |
| 41 | \( 1 + (457. - 134. i)T + (5.79e4 - 3.72e4i)T^{2} \) |
| 43 | \( 1 + (-1.21 + 8.42i)T + (-7.62e4 - 2.23e4i)T^{2} \) |
| 47 | \( 1 - 239.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (-49.4 - 57.1i)T + (-2.11e4 + 1.47e5i)T^{2} \) |
| 59 | \( 1 + (-136. + 157. i)T + (-2.92e4 - 2.03e5i)T^{2} \) |
| 61 | \( 1 + (45.8 + 318. i)T + (-2.17e5 + 6.39e4i)T^{2} \) |
| 67 | \( 1 + (583. - 375. i)T + (1.24e5 - 2.73e5i)T^{2} \) |
| 71 | \( 1 + (-76.1 + 48.9i)T + (1.48e5 - 3.25e5i)T^{2} \) |
| 73 | \( 1 + (-44.1 + 96.5i)T + (-2.54e5 - 2.93e5i)T^{2} \) |
| 79 | \( 1 + (373. - 430. i)T + (-7.01e4 - 4.88e5i)T^{2} \) |
| 83 | \( 1 + (713. + 209. i)T + (4.81e5 + 3.09e5i)T^{2} \) |
| 89 | \( 1 + (79.7 - 554. i)T + (-6.76e5 - 1.98e5i)T^{2} \) |
| 97 | \( 1 + (538. - 158. i)T + (7.67e5 - 4.93e5i)T^{2} \) |
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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
−17.18756106088494599731096401051, −16.84589229856499232754070347766, −14.55255527580054103212783734833, −13.35817050474939430536951234871, −12.35552077491752686357899851958, −9.869534417933488543010898976463, −8.795912708993995572419219098463, −7.10072357285910281103413289011, −6.52871288584274773499948194040, −1.39647407296357926090513752906,
2.95982300969810993228438341518, 5.66392178025128411109011058720, 8.862218792156276613401793248941, 9.600605217855692398170463931035, 10.32196200577914456358675281799, 11.68127190538685697764141531124, 14.00293191550847154878357230164, 15.18878926475722880345474034739, 16.65393784185072304419236931789, 17.45956818367491137832093911564
