| L(s) = 1 | + (−0.320 + 1.09i)2-s + (1.24 + 1.07i)3-s + (0.591 + 0.379i)4-s + (0.429 − 2.19i)5-s + (−1.57 + 1.01i)6-s + (1.40 − 0.643i)7-s + (−2.32 + 2.01i)8-s + (−0.0408 − 0.283i)9-s + (2.26 + 1.17i)10-s + (−4.00 + 1.17i)11-s + (0.326 + 1.11i)12-s + (−5.39 − 2.46i)13-s + (0.251 + 1.74i)14-s + (2.90 − 2.26i)15-s + (−0.872 − 1.91i)16-s + (0.798 + 1.24i)17-s + ⋯ |
| L(s) = 1 | + (−0.226 + 0.772i)2-s + (0.718 + 0.622i)3-s + (0.295 + 0.189i)4-s + (0.191 − 0.981i)5-s + (−0.644 + 0.414i)6-s + (0.532 − 0.243i)7-s + (−0.822 + 0.712i)8-s + (−0.0136 − 0.0946i)9-s + (0.714 + 0.371i)10-s + (−1.20 + 0.354i)11-s + (0.0941 + 0.320i)12-s + (−1.49 − 0.682i)13-s + (0.0671 + 0.466i)14-s + (0.749 − 0.585i)15-s + (−0.218 − 0.477i)16-s + (0.193 + 0.301i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.356 - 0.934i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.356 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.00855 + 0.694910i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.00855 + 0.694910i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-0.429 + 2.19i)T \) |
| 23 | \( 1 + (3.89 + 2.79i)T \) |
| good | 2 | \( 1 + (0.320 - 1.09i)T + (-1.68 - 1.08i)T^{2} \) |
| 3 | \( 1 + (-1.24 - 1.07i)T + (0.426 + 2.96i)T^{2} \) |
| 7 | \( 1 + (-1.40 + 0.643i)T + (4.58 - 5.29i)T^{2} \) |
| 11 | \( 1 + (4.00 - 1.17i)T + (9.25 - 5.94i)T^{2} \) |
| 13 | \( 1 + (5.39 + 2.46i)T + (8.51 + 9.82i)T^{2} \) |
| 17 | \( 1 + (-0.798 - 1.24i)T + (-7.06 + 15.4i)T^{2} \) |
| 19 | \( 1 + (-6.07 - 3.90i)T + (7.89 + 17.2i)T^{2} \) |
| 29 | \( 1 + (-2.24 + 1.44i)T + (12.0 - 26.3i)T^{2} \) |
| 31 | \( 1 + (-0.638 - 0.736i)T + (-4.41 + 30.6i)T^{2} \) |
| 37 | \( 1 + (-6.34 + 0.912i)T + (35.5 - 10.4i)T^{2} \) |
| 41 | \( 1 + (0.434 - 3.01i)T + (-39.3 - 11.5i)T^{2} \) |
| 43 | \( 1 + (-7.10 - 6.15i)T + (6.11 + 42.5i)T^{2} \) |
| 47 | \( 1 - 3.03iT - 47T^{2} \) |
| 53 | \( 1 + (-1.91 + 0.872i)T + (34.7 - 40.0i)T^{2} \) |
| 59 | \( 1 + (1.42 - 3.13i)T + (-38.6 - 44.5i)T^{2} \) |
| 61 | \( 1 + (1.61 + 1.86i)T + (-8.68 + 60.3i)T^{2} \) |
| 67 | \( 1 + (-1.00 + 3.43i)T + (-56.3 - 36.2i)T^{2} \) |
| 71 | \( 1 + (6.31 + 1.85i)T + (59.7 + 38.3i)T^{2} \) |
| 73 | \( 1 + (4.77 - 7.43i)T + (-30.3 - 66.4i)T^{2} \) |
| 79 | \( 1 + (-2.53 + 5.54i)T + (-51.7 - 59.7i)T^{2} \) |
| 83 | \( 1 + (-4.51 + 0.648i)T + (79.6 - 23.3i)T^{2} \) |
| 89 | \( 1 + (-5.73 + 6.62i)T + (-12.6 - 88.0i)T^{2} \) |
| 97 | \( 1 + (13.8 + 1.98i)T + (93.0 + 27.3i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.18384919140819502674419507507, −12.68692666086649194841062016393, −11.93302836381445860857015633051, −10.21532990580952489791637871143, −9.413296775647043250465297267466, −7.993154638014243778148952413814, −7.78589336839933491700741271416, −5.78077563292836539623145850746, −4.64168961457604877943466248105, −2.73554315080690952841954135385,
2.20670902564531152700368598496, 2.89463604794316583122315059184, 5.38189553417673938982689409552, 7.06573172013099519984975457620, 7.75448706986087971154532207294, 9.363957584845433685592284716454, 10.29693155015952255119597164251, 11.30916069692186702193946721702, 12.10805404383307145304231032048, 13.47888590752231376948761165350
