| L(s) = 1 | + (−0.923 + 2.02i)2-s + (−2.90 − 0.853i)3-s + (−1.92 − 2.22i)4-s + (0.841 − 0.540i)5-s + (4.40 − 5.08i)6-s + (0.283 − 1.97i)7-s + (2.00 − 0.589i)8-s + (5.19 + 3.34i)9-s + (0.316 + 2.20i)10-s + (−1.18 − 2.60i)11-s + (3.70 + 8.10i)12-s + (−0.658 − 4.58i)13-s + (3.72 + 2.39i)14-s + (−2.90 + 0.853i)15-s + (0.175 − 1.22i)16-s + (−2.52 + 2.91i)17-s + ⋯ |
| L(s) = 1 | + (−0.652 + 1.42i)2-s + (−1.67 − 0.492i)3-s + (−0.962 − 1.11i)4-s + (0.376 − 0.241i)5-s + (1.80 − 2.07i)6-s + (0.107 − 0.745i)7-s + (0.709 − 0.208i)8-s + (1.73 + 1.11i)9-s + (0.100 + 0.695i)10-s + (−0.358 − 0.784i)11-s + (1.06 + 2.33i)12-s + (−0.182 − 1.27i)13-s + (0.996 + 0.640i)14-s + (−0.750 + 0.220i)15-s + (0.0439 − 0.305i)16-s + (−0.611 + 0.705i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.738 + 0.674i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.738 + 0.674i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.293027 - 0.113649i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.293027 - 0.113649i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-0.841 + 0.540i)T \) |
| 23 | \( 1 + (-4.45 + 1.76i)T \) |
| good | 2 | \( 1 + (0.923 - 2.02i)T + (-1.30 - 1.51i)T^{2} \) |
| 3 | \( 1 + (2.90 + 0.853i)T + (2.52 + 1.62i)T^{2} \) |
| 7 | \( 1 + (-0.283 + 1.97i)T + (-6.71 - 1.97i)T^{2} \) |
| 11 | \( 1 + (1.18 + 2.60i)T + (-7.20 + 8.31i)T^{2} \) |
| 13 | \( 1 + (0.658 + 4.58i)T + (-12.4 + 3.66i)T^{2} \) |
| 17 | \( 1 + (2.52 - 2.91i)T + (-2.41 - 16.8i)T^{2} \) |
| 19 | \( 1 + (3.76 + 4.34i)T + (-2.70 + 18.8i)T^{2} \) |
| 29 | \( 1 + (0.374 - 0.431i)T + (-4.12 - 28.7i)T^{2} \) |
| 31 | \( 1 + (8.99 - 2.64i)T + (26.0 - 16.7i)T^{2} \) |
| 37 | \( 1 + (2.88 + 1.85i)T + (15.3 + 33.6i)T^{2} \) |
| 41 | \( 1 + (-0.228 + 0.146i)T + (17.0 - 37.2i)T^{2} \) |
| 43 | \( 1 + (-10.0 - 2.95i)T + (36.1 + 23.2i)T^{2} \) |
| 47 | \( 1 + 6.20T + 47T^{2} \) |
| 53 | \( 1 + (-0.378 + 2.63i)T + (-50.8 - 14.9i)T^{2} \) |
| 59 | \( 1 + (-0.571 - 3.97i)T + (-56.6 + 16.6i)T^{2} \) |
| 61 | \( 1 + (-6.61 + 1.94i)T + (51.3 - 32.9i)T^{2} \) |
| 67 | \( 1 + (-1.75 + 3.84i)T + (-43.8 - 50.6i)T^{2} \) |
| 71 | \( 1 + (-1.92 + 4.20i)T + (-46.4 - 53.6i)T^{2} \) |
| 73 | \( 1 + (7.00 + 8.08i)T + (-10.3 + 72.2i)T^{2} \) |
| 79 | \( 1 + (0.457 + 3.18i)T + (-75.7 + 22.2i)T^{2} \) |
| 83 | \( 1 + (-2.44 - 1.56i)T + (34.4 + 75.4i)T^{2} \) |
| 89 | \( 1 + (-4.06 - 1.19i)T + (74.8 + 48.1i)T^{2} \) |
| 97 | \( 1 + (-12.4 + 8.01i)T + (40.2 - 88.2i)T^{2} \) |
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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
−13.30688147908310288839729041143, −12.67454211035748731407626523608, −10.98384968148201316193439382196, −10.50926499122334056276615258634, −8.874835212178378479063769026366, −7.62260048375451278431825502954, −6.71215728604114334751585303632, −5.80976513424506334488670376863, −4.95207700834245212174744530968, −0.51476577491929584813250457777,
2.02720008921963865548012920787, 4.23691069455692536823781787837, 5.55898460377401297399481463154, 6.89173006893657863315139455138, 9.063390817445587372527334243174, 9.795354317701834409727895613287, 10.76455988921243346425329681910, 11.42729064847785954841719384517, 12.18498738116090028417345798922, 12.96016270195955785083031644141
