| L(s) = 1 | + (1.30 − 1.51i)2-s + (7.23 − 4.64i)3-s + (−0.569 − 3.95i)4-s + (5.71 + 12.5i)5-s + (2.44 − 17.0i)6-s + (−32.6 + 9.57i)7-s + (−6.73 − 4.32i)8-s + (19.5 − 42.7i)9-s + (26.4 + 7.75i)10-s + (18.7 + 21.6i)11-s + (−22.5 − 25.9i)12-s + (−30.5 − 8.96i)13-s + (−28.2 + 61.8i)14-s + (99.5 + 63.9i)15-s + (−15.3 + 4.50i)16-s + (11.0 − 77.1i)17-s + ⋯ |
| L(s) = 1 | + (0.463 − 0.534i)2-s + (1.39 − 0.894i)3-s + (−0.0711 − 0.494i)4-s + (0.511 + 1.11i)5-s + (0.166 − 1.15i)6-s + (−1.76 + 0.516i)7-s + (−0.297 − 0.191i)8-s + (0.722 − 1.58i)9-s + (0.835 + 0.245i)10-s + (0.513 + 0.592i)11-s + (−0.541 − 0.625i)12-s + (−0.651 − 0.191i)13-s + (−0.539 + 1.18i)14-s + (1.71 + 1.10i)15-s + (−0.239 + 0.0704i)16-s + (0.158 − 1.10i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.555 + 0.831i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.555 + 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.96407 - 1.04931i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.96407 - 1.04931i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.30 + 1.51i)T \) |
| 23 | \( 1 + (12.4 - 109. i)T \) |
| good | 3 | \( 1 + (-7.23 + 4.64i)T + (11.2 - 24.5i)T^{2} \) |
| 5 | \( 1 + (-5.71 - 12.5i)T + (-81.8 + 94.4i)T^{2} \) |
| 7 | \( 1 + (32.6 - 9.57i)T + (288. - 185. i)T^{2} \) |
| 11 | \( 1 + (-18.7 - 21.6i)T + (-189. + 1.31e3i)T^{2} \) |
| 13 | \( 1 + (30.5 + 8.96i)T + (1.84e3 + 1.18e3i)T^{2} \) |
| 17 | \( 1 + (-11.0 + 77.1i)T + (-4.71e3 - 1.38e3i)T^{2} \) |
| 19 | \( 1 + (-1.30 - 9.06i)T + (-6.58e3 + 1.93e3i)T^{2} \) |
| 29 | \( 1 + (-18.5 + 129. i)T + (-2.34e4 - 6.87e3i)T^{2} \) |
| 31 | \( 1 + (-170. - 109. i)T + (1.23e4 + 2.70e4i)T^{2} \) |
| 37 | \( 1 + (-123. + 270. i)T + (-3.31e4 - 3.82e4i)T^{2} \) |
| 41 | \( 1 + (7.14 + 15.6i)T + (-4.51e4 + 5.20e4i)T^{2} \) |
| 43 | \( 1 + (1.67 - 1.07i)T + (3.30e4 - 7.23e4i)T^{2} \) |
| 47 | \( 1 + 341.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (472. - 138. i)T + (1.25e5 - 8.04e4i)T^{2} \) |
| 59 | \( 1 + (-124. - 36.4i)T + (1.72e5 + 1.11e5i)T^{2} \) |
| 61 | \( 1 + (170. + 109. i)T + (9.42e4 + 2.06e5i)T^{2} \) |
| 67 | \( 1 + (472. - 545. i)T + (-4.28e4 - 2.97e5i)T^{2} \) |
| 71 | \( 1 + (-495. + 571. i)T + (-5.09e4 - 3.54e5i)T^{2} \) |
| 73 | \( 1 + (-90.1 - 627. i)T + (-3.73e5 + 1.09e5i)T^{2} \) |
| 79 | \( 1 + (608. + 178. i)T + (4.14e5 + 2.66e5i)T^{2} \) |
| 83 | \( 1 + (-289. + 634. i)T + (-3.74e5 - 4.32e5i)T^{2} \) |
| 89 | \( 1 + (66.5 - 42.7i)T + (2.92e5 - 6.41e5i)T^{2} \) |
| 97 | \( 1 + (-304. - 666. i)T + (-5.97e5 + 6.89e5i)T^{2} \) |
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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
−14.67177263750928356829953035466, −13.86550946028240713273597598722, −12.98643119359151895152291656249, −12.01467440577974247963001897575, −9.915254228436576602603001538892, −9.353354974562201936881221512241, −7.27789467726539306062655289850, −6.32655038396852035822063595186, −3.24477255606287333689807756204, −2.44920331431484318497001480031,
3.20142276323232237924712402759, 4.47314676727892422413630590651, 6.38695681065832687227925195225, 8.303502061462537105014899861670, 9.282099808414087192539045037644, 10.07589438145174132891070153631, 12.62585173803839286375969121997, 13.36057828530966172652359582783, 14.29925123026406559132605674774, 15.44408913333558200895376467194
