| L(s) = 1 | + (0.173 + 0.984i)2-s + (−1.16 + 0.978i)3-s + (−0.939 + 0.342i)4-s + (2.75 + 1.00i)5-s + (−1.16 − 0.978i)6-s + (0.232 − 0.402i)7-s + (−0.5 − 0.866i)8-s + (−0.118 + 0.674i)9-s + (−0.509 + 2.89i)10-s + (0.5 + 0.866i)11-s + (0.760 − 1.31i)12-s + (3.44 + 2.89i)13-s + (0.436 + 0.158i)14-s + (−4.19 + 1.52i)15-s + (0.766 − 0.642i)16-s + (0.0327 + 0.185i)17-s + ⋯ |
| L(s) = 1 | + (0.122 + 0.696i)2-s + (−0.672 + 0.564i)3-s + (−0.469 + 0.171i)4-s + (1.23 + 0.449i)5-s + (−0.475 − 0.399i)6-s + (0.0878 − 0.152i)7-s + (−0.176 − 0.306i)8-s + (−0.0396 + 0.224i)9-s + (−0.161 + 0.914i)10-s + (0.150 + 0.261i)11-s + (0.219 − 0.380i)12-s + (0.955 + 0.802i)13-s + (0.116 + 0.0424i)14-s + (−1.08 + 0.394i)15-s + (0.191 − 0.160i)16-s + (0.00795 + 0.0450i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.708 - 0.705i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.708 - 0.705i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.502745 + 1.21831i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.502745 + 1.21831i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.173 - 0.984i)T \) |
| 11 | \( 1 + (-0.5 - 0.866i)T \) |
| 19 | \( 1 + (4.22 - 1.07i)T \) |
| good | 3 | \( 1 + (1.16 - 0.978i)T + (0.520 - 2.95i)T^{2} \) |
| 5 | \( 1 + (-2.75 - 1.00i)T + (3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (-0.232 + 0.402i)T + (-3.5 - 6.06i)T^{2} \) |
| 13 | \( 1 + (-3.44 - 2.89i)T + (2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-0.0327 - 0.185i)T + (-15.9 + 5.81i)T^{2} \) |
| 23 | \( 1 + (2.73 - 0.995i)T + (17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (-0.299 + 1.69i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (-0.937 + 1.62i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 6.48T + 37T^{2} \) |
| 41 | \( 1 + (2.52 - 2.12i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-10.0 - 3.66i)T + (32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (0.699 - 3.96i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + (-1.27 + 0.464i)T + (40.6 - 34.0i)T^{2} \) |
| 59 | \( 1 + (0.956 + 5.42i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (-10.3 + 3.75i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (-0.966 + 5.48i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (3.74 + 1.36i)T + (54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (-8.35 + 7.00i)T + (12.6 - 71.8i)T^{2} \) |
| 79 | \( 1 + (-4.24 + 3.56i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-6.42 + 11.1i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-12.9 - 10.8i)T + (15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (2.10 + 11.9i)T + (-91.1 + 33.1i)T^{2} \) |
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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
−11.30082946002359839501180042663, −10.55327590953340704273151581179, −9.806362561274183656158788902738, −8.919164250393505611792873118451, −7.76059329435817226625316973199, −6.37872868850885832608884348925, −6.08440702748710226386201698598, −4.94869134201398743675740296751, −3.94332402805330696448205131086, −2.03966471790663431494125815757,
0.942485934666305077539323056862, 2.18203320351901214669296349617, 3.75689714434171944235369732432, 5.33102173213942040255226929259, 5.87353957419244393752150706841, 6.79745238165632458121946821314, 8.454485494106477930928235585852, 9.077359994147167328624736016358, 10.20857595691996260854739200137, 10.84743801876994468285983080577
