| L(s) = 1 | + (−2.10 − 4.61i)2-s + (8.43 − 2.47i)3-s + (−11.6 + 13.4i)4-s + (−5.02 − 3.22i)5-s + (−29.1 − 33.6i)6-s + (3.12 + 21.7i)7-s + (47.3 + 13.9i)8-s + (42.2 − 27.1i)9-s + (−4.31 + 29.9i)10-s + (4.09 − 8.97i)11-s + (−64.7 + 141. i)12-s + (2.55 − 17.7i)13-s + (93.5 − 60.1i)14-s + (−50.3 − 14.7i)15-s + (−15.4 − 107. i)16-s + (17.5 + 20.2i)17-s + ⋯ |
| L(s) = 1 | + (−0.745 − 1.63i)2-s + (1.62 − 0.476i)3-s + (−1.45 + 1.67i)4-s + (−0.449 − 0.288i)5-s + (−1.98 − 2.29i)6-s + (0.168 + 1.17i)7-s + (2.09 + 0.614i)8-s + (1.56 − 1.00i)9-s + (−0.136 + 0.948i)10-s + (0.112 − 0.245i)11-s + (−1.55 + 3.41i)12-s + (0.0545 − 0.379i)13-s + (1.78 − 1.14i)14-s + (−0.866 − 0.254i)15-s + (−0.241 − 1.67i)16-s + (0.250 + 0.288i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.443 + 0.896i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.443 + 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.579333 - 0.933003i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.579333 - 0.933003i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 + (93.1 - 59.0i)T \) |
| good | 2 | \( 1 + (2.10 + 4.61i)T + (-5.23 + 6.04i)T^{2} \) |
| 3 | \( 1 + (-8.43 + 2.47i)T + (22.7 - 14.5i)T^{2} \) |
| 5 | \( 1 + (5.02 + 3.22i)T + (51.9 + 113. i)T^{2} \) |
| 7 | \( 1 + (-3.12 - 21.7i)T + (-329. + 96.6i)T^{2} \) |
| 11 | \( 1 + (-4.09 + 8.97i)T + (-871. - 1.00e3i)T^{2} \) |
| 13 | \( 1 + (-2.55 + 17.7i)T + (-2.10e3 - 618. i)T^{2} \) |
| 17 | \( 1 + (-17.5 - 20.2i)T + (-699. + 4.86e3i)T^{2} \) |
| 19 | \( 1 + (45.4 - 52.4i)T + (-976. - 6.78e3i)T^{2} \) |
| 29 | \( 1 + (-26.8 - 30.9i)T + (-3.47e3 + 2.41e4i)T^{2} \) |
| 31 | \( 1 + (-85.9 - 25.2i)T + (2.50e4 + 1.61e4i)T^{2} \) |
| 37 | \( 1 + (372. - 239. i)T + (2.10e4 - 4.60e4i)T^{2} \) |
| 41 | \( 1 + (238. + 153. i)T + (2.86e4 + 6.26e4i)T^{2} \) |
| 43 | \( 1 + (-510. + 149. i)T + (6.68e4 - 4.29e4i)T^{2} \) |
| 47 | \( 1 + 228.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (52.9 + 368. i)T + (-1.42e5 + 4.19e4i)T^{2} \) |
| 59 | \( 1 + (-105. + 733. i)T + (-1.97e5 - 5.78e4i)T^{2} \) |
| 61 | \( 1 + (-504. - 148. i)T + (1.90e5 + 1.22e5i)T^{2} \) |
| 67 | \( 1 + (18.7 + 41.1i)T + (-1.96e5 + 2.27e5i)T^{2} \) |
| 71 | \( 1 + (-74.1 - 162. i)T + (-2.34e5 + 2.70e5i)T^{2} \) |
| 73 | \( 1 + (147. - 169. i)T + (-5.53e4 - 3.85e5i)T^{2} \) |
| 79 | \( 1 + (-8.57 + 59.6i)T + (-4.73e5 - 1.38e5i)T^{2} \) |
| 83 | \( 1 + (-387. + 249. i)T + (2.37e5 - 5.20e5i)T^{2} \) |
| 89 | \( 1 + (82.8 - 24.3i)T + (5.93e5 - 3.81e5i)T^{2} \) |
| 97 | \( 1 + (522. + 336. i)T + (3.79e5 + 8.30e5i)T^{2} \) |
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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
−17.73312477097853542058061962391, −15.55588942011457741689571120363, −14.06089106155370688084870848174, −12.70507195622483896982323332056, −11.92237487882613013388282207574, −10.00811591432357018126053871664, −8.662386175484583849948128178792, −8.185720002556080324588976762228, −3.53623728399808708063776435630, −2.03013856270085475937654386551,
4.23454028291830758286295145909, 7.06913104143960364593637498943, 7.970859858780554051829744893331, 9.150932529947276956982769199195, 10.34228008870615584607271993483, 13.67325782539776515230966208273, 14.38174325872598698213441594564, 15.31953508941960195242133196676, 16.25373772136673683999710124547, 17.52308135440818955989818365097
