| L(s) = 1 | − 1.29·3-s + (−0.437 − 2.19i)5-s − i·7-s − 1.32·9-s + 0.711i·11-s − 1.85·13-s + (0.564 + 2.83i)15-s − 3.75i·17-s − 7.57i·19-s + 1.29i·21-s + 8.45i·23-s + (−4.61 + 1.91i)25-s + 5.59·27-s − 5.24i·29-s + 0.349·31-s + ⋯ |
| L(s) = 1 | − 0.746·3-s + (−0.195 − 0.980i)5-s − 0.377i·7-s − 0.443·9-s + 0.214i·11-s − 0.514·13-s + (0.145 + 0.731i)15-s − 0.911i·17-s − 1.73i·19-s + 0.282i·21-s + 1.76i·23-s + (−0.923 + 0.383i)25-s + 1.07·27-s − 0.974i·29-s + 0.0626·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.442 - 0.896i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.442 - 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.03127485732\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.03127485732\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.437 + 2.19i)T \) |
| 7 | \( 1 + iT \) |
| good | 3 | \( 1 + 1.29T + 3T^{2} \) |
| 11 | \( 1 - 0.711iT - 11T^{2} \) |
| 13 | \( 1 + 1.85T + 13T^{2} \) |
| 17 | \( 1 + 3.75iT - 17T^{2} \) |
| 19 | \( 1 + 7.57iT - 19T^{2} \) |
| 23 | \( 1 - 8.45iT - 23T^{2} \) |
| 29 | \( 1 + 5.24iT - 29T^{2} \) |
| 31 | \( 1 - 0.349T + 31T^{2} \) |
| 37 | \( 1 + 7.50T + 37T^{2} \) |
| 41 | \( 1 - 2.74T + 41T^{2} \) |
| 43 | \( 1 + 2.07T + 43T^{2} \) |
| 47 | \( 1 - 4.69iT - 47T^{2} \) |
| 53 | \( 1 + 9.81T + 53T^{2} \) |
| 59 | \( 1 - 6.54iT - 59T^{2} \) |
| 61 | \( 1 + 6.50iT - 61T^{2} \) |
| 67 | \( 1 - 10.4T + 67T^{2} \) |
| 71 | \( 1 - 5.53T + 71T^{2} \) |
| 73 | \( 1 - 9.10iT - 73T^{2} \) |
| 79 | \( 1 + 4.54T + 79T^{2} \) |
| 83 | \( 1 - 4.92T + 83T^{2} \) |
| 89 | \( 1 + 5.58T + 89T^{2} \) |
| 97 | \( 1 - 14.1iT - 97T^{2} \) |
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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
−9.391142014582241432694491799276, −8.567642854219249765681093028097, −7.64702094719145528133976024296, −7.02535570178209738762357261936, −6.04687481508847607544114095581, −5.00505848964839766730397300257, −4.93061585061963280967786632937, −3.67424149099831365378388472425, −2.48525315887273706429714596847, −1.02837974998123553887520142417,
0.01375695228333790945296441215, 1.86073665235427599189042194914, 2.95330394924869168784074271940, 3.79127607325849913296122624412, 4.92236775464574479076469077248, 5.81074216289313924690979005969, 6.34375246574261487052080784336, 7.03830165165378562982095604225, 8.147830359198170683966206588672, 8.544344449774950523068354696056
