| L(s) = 1 | + (−1.80 − 0.528i)2-s + (2.35 − 2.72i)3-s + (−0.402 − 0.258i)4-s + (5.05 + 0.726i)5-s + (−5.68 + 3.65i)6-s + (−8.85 + 4.04i)7-s + (5.50 + 6.35i)8-s + (−0.563 − 3.92i)9-s + (−8.71 − 3.97i)10-s + (2.47 + 8.43i)11-s + (−1.65 + 0.485i)12-s + (5.88 − 12.8i)13-s + (18.0 − 2.59i)14-s + (13.8 − 12.0i)15-s + (−5.75 − 12.6i)16-s + (−3.89 − 6.05i)17-s + ⋯ |
| L(s) = 1 | + (−0.900 − 0.264i)2-s + (0.785 − 0.906i)3-s + (−0.100 − 0.0646i)4-s + (1.01 + 0.145i)5-s + (−0.947 + 0.608i)6-s + (−1.26 + 0.577i)7-s + (0.687 + 0.793i)8-s + (−0.0626 − 0.435i)9-s + (−0.871 − 0.397i)10-s + (0.225 + 0.766i)11-s + (−0.137 + 0.0404i)12-s + (0.453 − 0.992i)13-s + (1.29 − 0.185i)14-s + (0.925 − 0.802i)15-s + (−0.359 − 0.787i)16-s + (−0.228 − 0.356i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.622 + 0.782i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.622 + 0.782i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.655999 - 0.316430i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.655999 - 0.316430i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 + (11.2 + 20.0i)T \) |
| good | 2 | \( 1 + (1.80 + 0.528i)T + (3.36 + 2.16i)T^{2} \) |
| 3 | \( 1 + (-2.35 + 2.72i)T + (-1.28 - 8.90i)T^{2} \) |
| 5 | \( 1 + (-5.05 - 0.726i)T + (23.9 + 7.04i)T^{2} \) |
| 7 | \( 1 + (8.85 - 4.04i)T + (32.0 - 37.0i)T^{2} \) |
| 11 | \( 1 + (-2.47 - 8.43i)T + (-101. + 65.4i)T^{2} \) |
| 13 | \( 1 + (-5.88 + 12.8i)T + (-110. - 127. i)T^{2} \) |
| 17 | \( 1 + (3.89 + 6.05i)T + (-120. + 262. i)T^{2} \) |
| 19 | \( 1 + (15.1 - 23.5i)T + (-149. - 328. i)T^{2} \) |
| 29 | \( 1 + (-10.1 + 6.53i)T + (349. - 765. i)T^{2} \) |
| 31 | \( 1 + (28.5 + 33.0i)T + (-136. + 951. i)T^{2} \) |
| 37 | \( 1 + (-32.2 + 4.64i)T + (1.31e3 - 385. i)T^{2} \) |
| 41 | \( 1 + (8.24 - 57.3i)T + (-1.61e3 - 473. i)T^{2} \) |
| 43 | \( 1 + (6.69 + 5.80i)T + (263. + 1.83e3i)T^{2} \) |
| 47 | \( 1 - 10.4T + 2.20e3T^{2} \) |
| 53 | \( 1 + (-20.7 + 9.47i)T + (1.83e3 - 2.12e3i)T^{2} \) |
| 59 | \( 1 + (3.35 - 7.34i)T + (-2.27e3 - 2.63e3i)T^{2} \) |
| 61 | \( 1 + (13.7 - 11.9i)T + (529. - 3.68e3i)T^{2} \) |
| 67 | \( 1 + (-21.0 + 71.8i)T + (-3.77e3 - 2.42e3i)T^{2} \) |
| 71 | \( 1 + (-48.3 - 14.2i)T + (4.24e3 + 2.72e3i)T^{2} \) |
| 73 | \( 1 + (61.7 + 39.6i)T + (2.21e3 + 4.84e3i)T^{2} \) |
| 79 | \( 1 + (-56.1 - 25.6i)T + (4.08e3 + 4.71e3i)T^{2} \) |
| 83 | \( 1 + (-75.3 + 10.8i)T + (6.60e3 - 1.94e3i)T^{2} \) |
| 89 | \( 1 + (-44.2 - 38.3i)T + (1.12e3 + 7.84e3i)T^{2} \) |
| 97 | \( 1 + (-8.57 - 1.23i)T + (9.02e3 + 2.65e3i)T^{2} \) |
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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
−18.07119280168235497521380813202, −16.63802068256688144126378529306, −14.74860448443699443343844013003, −13.49444329547319297878175829687, −12.65476840406499913034258860338, −10.26951139093920988120456206982, −9.360201152705622333750371558854, −8.038592810577284235958890652924, −6.16852793794585659097726268041, −2.23517194068558432823401597728,
3.82398591001900582092916963114, 6.63036115670050442036208586432, 8.861834134248371969893479674829, 9.398992014766000359357446323822, 10.48153350953569057484091636453, 13.18600554104263312729365745539, 13.96124861552244661117672895214, 15.79289143975743091969442792753, 16.59786811351257284805523999797, 17.64501473586825050572171577547
