| L(s) = 1 | − 1.86i·3-s + (2.07 + 0.836i)5-s + (−2.64 + 0.168i)7-s − 0.486·9-s + 4.26i·11-s − 5.57·13-s + (1.56 − 3.87i)15-s + 3.01·17-s + 0.0464·19-s + (0.314 + 4.93i)21-s − 5.58·23-s + (3.60 + 3.46i)25-s − 4.69i·27-s − 1.21·29-s + 4.62·31-s + ⋯ |
| L(s) = 1 | − 1.07i·3-s + (0.927 + 0.374i)5-s + (−0.997 + 0.0635i)7-s − 0.162·9-s + 1.28i·11-s − 1.54·13-s + (0.403 − 0.999i)15-s + 0.731·17-s + 0.0106·19-s + (0.0685 + 1.07i)21-s − 1.16·23-s + (0.720 + 0.693i)25-s − 0.903i·27-s − 0.224·29-s + 0.829·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.314 - 0.949i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.314 - 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.131396529\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.131396529\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-2.07 - 0.836i)T \) |
| 7 | \( 1 + (2.64 - 0.168i)T \) |
| good | 3 | \( 1 + 1.86iT - 3T^{2} \) |
| 11 | \( 1 - 4.26iT - 11T^{2} \) |
| 13 | \( 1 + 5.57T + 13T^{2} \) |
| 17 | \( 1 - 3.01T + 17T^{2} \) |
| 19 | \( 1 - 0.0464T + 19T^{2} \) |
| 23 | \( 1 + 5.58T + 23T^{2} \) |
| 29 | \( 1 + 1.21T + 29T^{2} \) |
| 31 | \( 1 - 4.62T + 31T^{2} \) |
| 37 | \( 1 - 9.87iT - 37T^{2} \) |
| 41 | \( 1 - 10.5iT - 41T^{2} \) |
| 43 | \( 1 + 12.4T + 43T^{2} \) |
| 47 | \( 1 - 6.78iT - 47T^{2} \) |
| 53 | \( 1 + 4.99iT - 53T^{2} \) |
| 59 | \( 1 + 7.06T + 59T^{2} \) |
| 61 | \( 1 - 11.5iT - 61T^{2} \) |
| 67 | \( 1 - 6.65T + 67T^{2} \) |
| 71 | \( 1 + 5.11iT - 71T^{2} \) |
| 73 | \( 1 - 9.67T + 73T^{2} \) |
| 79 | \( 1 - 5.90iT - 79T^{2} \) |
| 83 | \( 1 - 11.1iT - 83T^{2} \) |
| 89 | \( 1 - 1.51iT - 89T^{2} \) |
| 97 | \( 1 - 3.01T + 97T^{2} \) |
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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
−9.651164519512878831098908841510, −8.144500808050008470483215125361, −7.52697071600458365160385421590, −6.60103710485404503542820400891, −6.54882325446563039046743333107, −5.36103398747474594748475848800, −4.48196488017513489404102310343, −3.02454953351083531484998044753, −2.31443852169637375543002114141, −1.42307052049147972406989012583,
0.37016728708373162815321559184, 2.11701257005444519579089619674, 3.19632759961284619830660973685, 3.92279894141991354857728699737, 5.01896108215962983518732694731, 5.57344086796227518732242436998, 6.34975415961428424908292341177, 7.28828165024651765520186559959, 8.336292671616450700767387475539, 9.210569197099689282236308653402
