| L(s) = 1 | + (0.0260 − 0.297i)2-s + (−0.818 − 1.75i)3-s + (1.88 + 0.331i)4-s + (−1.72 − 1.41i)5-s + (−0.544 + 0.198i)6-s + (−0.323 + 0.0866i)7-s + (0.302 − 1.12i)8-s + (−0.485 + 0.578i)9-s + (−0.467 + 0.477i)10-s + (0.885 + 1.53i)11-s + (−0.958 − 3.57i)12-s + (2.07 + 0.968i)13-s + (0.0173 + 0.0984i)14-s + (−1.07 + 4.19i)15-s + (3.26 + 1.18i)16-s + (−2.73 − 0.238i)17-s + ⋯ |
| L(s) = 1 | + (0.0184 − 0.210i)2-s + (−0.472 − 1.01i)3-s + (0.940 + 0.165i)4-s + (−0.772 − 0.635i)5-s + (−0.222 + 0.0808i)6-s + (−0.122 + 0.0327i)7-s + (0.106 − 0.399i)8-s + (−0.161 + 0.192i)9-s + (−0.147 + 0.150i)10-s + (0.266 + 0.462i)11-s + (−0.276 − 1.03i)12-s + (0.576 + 0.268i)13-s + (0.00464 + 0.0263i)14-s + (−0.278 + 1.08i)15-s + (0.815 + 0.296i)16-s + (−0.662 − 0.0579i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.251 + 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.251 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.762475 - 0.589840i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.762475 - 0.589840i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (1.72 + 1.41i)T \) |
| 19 | \( 1 + (-0.174 - 4.35i)T \) |
| good | 2 | \( 1 + (-0.0260 + 0.297i)T + (-1.96 - 0.347i)T^{2} \) |
| 3 | \( 1 + (0.818 + 1.75i)T + (-1.92 + 2.29i)T^{2} \) |
| 7 | \( 1 + (0.323 - 0.0866i)T + (6.06 - 3.5i)T^{2} \) |
| 11 | \( 1 + (-0.885 - 1.53i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.07 - 0.968i)T + (8.35 + 9.95i)T^{2} \) |
| 17 | \( 1 + (2.73 + 0.238i)T + (16.7 + 2.95i)T^{2} \) |
| 23 | \( 1 + (-4.93 - 7.04i)T + (-7.86 + 21.6i)T^{2} \) |
| 29 | \( 1 + (4.70 + 3.94i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (1.80 + 1.04i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-4.27 + 4.27i)T - 37iT^{2} \) |
| 41 | \( 1 + (-2.08 + 5.72i)T + (-31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (4.37 + 3.06i)T + (14.7 + 40.4i)T^{2} \) |
| 47 | \( 1 + (-0.503 - 5.75i)T + (-46.2 + 8.16i)T^{2} \) |
| 53 | \( 1 + (-1.74 + 1.22i)T + (18.1 - 49.8i)T^{2} \) |
| 59 | \( 1 + (-1.26 + 1.06i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (-1.59 + 9.05i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (10.9 - 0.957i)T + (65.9 - 11.6i)T^{2} \) |
| 71 | \( 1 + (15.3 - 2.71i)T + (66.7 - 24.2i)T^{2} \) |
| 73 | \( 1 + (5.01 - 2.33i)T + (46.9 - 55.9i)T^{2} \) |
| 79 | \( 1 + (-14.0 - 5.11i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (-0.859 - 3.20i)T + (-71.8 + 41.5i)T^{2} \) |
| 89 | \( 1 + (9.75 - 3.55i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (0.522 - 5.97i)T + (-95.5 - 16.8i)T^{2} \) |
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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
−13.27149831196691177396490876032, −12.57682466222889301246557455926, −11.72472854835995920867324644838, −11.10970107620790025140772088651, −9.362883332609314477636949504218, −7.79943843460739039917563954899, −7.06207129165697505961716509594, −5.85664400556543711390771611707, −3.82187376605865136045414499628, −1.57891181097718947387899282412,
3.12241586730126556548843870397, 4.67501614276341795875270422023, 6.22203956624396809803731677894, 7.23536654458663372996704698894, 8.716978796406491346696375387856, 10.33476069305052315799011072815, 11.02969211328752639294998302627, 11.56122445936052673654961403696, 13.15130678331004414417025335830, 14.86474561965535029275397962377
