| L(s) = 1 | + (0.0814 + 1.13i)2-s + (4.76 + 1.03i)3-s + (2.66 − 0.383i)4-s + (−4.76 + 1.52i)5-s + (−0.793 + 5.51i)6-s + (0.510 − 0.934i)7-s + (1.62 + 7.47i)8-s + (13.4 + 6.15i)9-s + (−2.12 − 5.30i)10-s + (−9.87 − 11.3i)11-s + (13.1 + 0.938i)12-s + (−6.65 − 12.1i)13-s + (1.10 + 0.505i)14-s + (−24.2 + 2.34i)15-s + (1.95 − 0.575i)16-s + (5.95 + 7.94i)17-s + ⋯ |
| L(s) = 1 | + (0.0407 + 0.569i)2-s + (1.58 + 0.345i)3-s + (0.666 − 0.0958i)4-s + (−0.952 + 0.305i)5-s + (−0.132 + 0.919i)6-s + (0.0728 − 0.133i)7-s + (0.203 + 0.934i)8-s + (1.49 + 0.683i)9-s + (−0.212 − 0.530i)10-s + (−0.897 − 1.03i)11-s + (1.09 + 0.0781i)12-s + (−0.511 − 0.937i)13-s + (0.0789 + 0.0360i)14-s + (−1.61 + 0.156i)15-s + (0.122 − 0.0359i)16-s + (0.350 + 0.467i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.547 - 0.836i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.547 - 0.836i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.93770 + 1.04802i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.93770 + 1.04802i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (4.76 - 1.52i)T \) |
| 23 | \( 1 + (-11.6 - 19.8i)T \) |
| good | 2 | \( 1 + (-0.0814 - 1.13i)T + (-3.95 + 0.569i)T^{2} \) |
| 3 | \( 1 + (-4.76 - 1.03i)T + (8.18 + 3.73i)T^{2} \) |
| 7 | \( 1 + (-0.510 + 0.934i)T + (-26.4 - 41.2i)T^{2} \) |
| 11 | \( 1 + (9.87 + 11.3i)T + (-17.2 + 119. i)T^{2} \) |
| 13 | \( 1 + (6.65 + 12.1i)T + (-91.3 + 142. i)T^{2} \) |
| 17 | \( 1 + (-5.95 - 7.94i)T + (-81.4 + 277. i)T^{2} \) |
| 19 | \( 1 + (3.51 - 0.505i)T + (346. - 101. i)T^{2} \) |
| 29 | \( 1 + (39.6 + 5.70i)T + (806. + 236. i)T^{2} \) |
| 31 | \( 1 + (7.80 + 5.01i)T + (399. + 874. i)T^{2} \) |
| 37 | \( 1 + (15.4 + 41.3i)T + (-1.03e3 + 896. i)T^{2} \) |
| 41 | \( 1 + (-3.94 - 8.64i)T + (-1.10e3 + 1.27e3i)T^{2} \) |
| 43 | \( 1 + (-26.7 - 5.81i)T + (1.68e3 + 768. i)T^{2} \) |
| 47 | \( 1 + (-53.5 + 53.5i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (51.3 + 28.0i)T + (1.51e3 + 2.36e3i)T^{2} \) |
| 59 | \( 1 + (28.2 - 96.0i)T + (-2.92e3 - 1.88e3i)T^{2} \) |
| 61 | \( 1 + (-1.55 - 0.999i)T + (1.54e3 + 3.38e3i)T^{2} \) |
| 67 | \( 1 + (0.842 + 11.7i)T + (-4.44e3 + 638. i)T^{2} \) |
| 71 | \( 1 + (-27.4 + 31.6i)T + (-717. - 4.98e3i)T^{2} \) |
| 73 | \( 1 + (-12.4 + 16.6i)T + (-1.50e3 - 5.11e3i)T^{2} \) |
| 79 | \( 1 + (32.5 - 110. i)T + (-5.25e3 - 3.37e3i)T^{2} \) |
| 83 | \( 1 + (129. - 48.4i)T + (5.20e3 - 4.51e3i)T^{2} \) |
| 89 | \( 1 + (12.6 + 19.7i)T + (-3.29e3 + 7.20e3i)T^{2} \) |
| 97 | \( 1 + (-141. - 52.8i)T + (7.11e3 + 6.16e3i)T^{2} \) |
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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
−13.85738077623612116940722561618, −12.70424940324664595824282835329, −11.17272013873262328561245500938, −10.38151255152549420664524470607, −8.831664918524784641479960284438, −7.78184107757271656176622796521, −7.49695728115278195752878438744, −5.55364815479086020265481738438, −3.66107642653554434761066266181, −2.64248624298208593575107115758,
1.98661286577135990425686121600, 3.11539690151081122995228472269, 4.46879335533267628778904080626, 7.13926843799651673173869183574, 7.59412674099527677551161725594, 8.827542052819160262416563560964, 9.883515448429996441253524537319, 11.21639333104567781509402898609, 12.41602926077320445277448443446, 12.84156179216417099658080516924
