| L(s) = 1 | + (−1.12 + 0.330i)2-s + (−6.34 − 7.32i)3-s + (−5.57 + 3.58i)4-s + (−0.0835 − 0.581i)5-s + (9.57 + 6.15i)6-s + (10.1 − 22.2i)7-s + (11.2 − 12.9i)8-s + (−9.53 + 66.2i)9-s + (0.286 + 0.627i)10-s + (−29.4 − 8.63i)11-s + (61.5 + 18.0i)12-s + (−11.4 − 25.1i)13-s + (−4.08 + 28.4i)14-s + (−3.72 + 4.30i)15-s + (13.6 − 29.8i)16-s + (−25.9 − 16.6i)17-s + ⋯ |
| L(s) = 1 | + (−0.398 + 0.116i)2-s + (−1.22 − 1.40i)3-s + (−0.696 + 0.447i)4-s + (−0.00747 − 0.0519i)5-s + (0.651 + 0.418i)6-s + (0.548 − 1.20i)7-s + (0.496 − 0.573i)8-s + (−0.353 + 2.45i)9-s + (0.00905 + 0.0198i)10-s + (−0.806 − 0.236i)11-s + (1.48 + 0.435i)12-s + (−0.244 − 0.536i)13-s + (−0.0780 + 0.542i)14-s + (−0.0641 + 0.0740i)15-s + (0.213 − 0.466i)16-s + (−0.369 − 0.237i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.748 + 0.663i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.748 + 0.663i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.151486 - 0.399100i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.151486 - 0.399100i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 + (69.3 - 85.8i)T \) |
| good | 2 | \( 1 + (1.12 - 0.330i)T + (6.73 - 4.32i)T^{2} \) |
| 3 | \( 1 + (6.34 + 7.32i)T + (-3.84 + 26.7i)T^{2} \) |
| 5 | \( 1 + (0.0835 + 0.581i)T + (-119. + 35.2i)T^{2} \) |
| 7 | \( 1 + (-10.1 + 22.2i)T + (-224. - 259. i)T^{2} \) |
| 11 | \( 1 + (29.4 + 8.63i)T + (1.11e3 + 719. i)T^{2} \) |
| 13 | \( 1 + (11.4 + 25.1i)T + (-1.43e3 + 1.66e3i)T^{2} \) |
| 17 | \( 1 + (25.9 + 16.6i)T + (2.04e3 + 4.46e3i)T^{2} \) |
| 19 | \( 1 + (-77.8 + 50.0i)T + (2.84e3 - 6.23e3i)T^{2} \) |
| 29 | \( 1 + (61.9 + 39.8i)T + (1.01e4 + 2.21e4i)T^{2} \) |
| 31 | \( 1 + (-68.9 + 79.5i)T + (-4.23e3 - 2.94e4i)T^{2} \) |
| 37 | \( 1 + (12.7 - 88.8i)T + (-4.86e4 - 1.42e4i)T^{2} \) |
| 41 | \( 1 + (-12.6 - 87.8i)T + (-6.61e4 + 1.94e4i)T^{2} \) |
| 43 | \( 1 + (239. + 276. i)T + (-1.13e4 + 7.86e4i)T^{2} \) |
| 47 | \( 1 - 181.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (-254. + 557. i)T + (-9.74e4 - 1.12e5i)T^{2} \) |
| 59 | \( 1 + (-32.4 - 70.9i)T + (-1.34e5 + 1.55e5i)T^{2} \) |
| 61 | \( 1 + (73.2 - 84.5i)T + (-3.23e4 - 2.24e5i)T^{2} \) |
| 67 | \( 1 + (-392. + 115. i)T + (2.53e5 - 1.62e5i)T^{2} \) |
| 71 | \( 1 + (431. - 126. i)T + (3.01e5 - 1.93e5i)T^{2} \) |
| 73 | \( 1 + (78.5 - 50.5i)T + (1.61e5 - 3.53e5i)T^{2} \) |
| 79 | \( 1 + (115. + 252. i)T + (-3.22e5 + 3.72e5i)T^{2} \) |
| 83 | \( 1 + (-158. + 1.10e3i)T + (-5.48e5 - 1.61e5i)T^{2} \) |
| 89 | \( 1 + (-833. - 961. i)T + (-1.00e5 + 6.97e5i)T^{2} \) |
| 97 | \( 1 + (-154. - 1.07e3i)T + (-8.75e5 + 2.57e5i)T^{2} \) |
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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
−17.34252486317108399116483036067, −16.31805554622507592876231855668, −13.68259097280157075102735233644, −13.15161847673330688950083070531, −11.72140350259379505180627020359, −10.41406508858431209432007363477, −7.984678463380708556265481125977, −7.16699921041925299703683039136, −5.08624033347442887754790516489, −0.61704732383762362164608402594,
4.70654344486063204771472745879, 5.68225303367842417660078673973, 8.800339460378438395566256420943, 9.940432818584984406051160896961, 10.99116441675045568165038800525, 12.23443273239288067118873208992, 14.52221102556902756461205472731, 15.50342303743175903313884478235, 16.66537612021542723955445989768, 17.92339179680440850415637892667
