| L(s) = 1 | + i·5-s + (−0.321 − 2.62i)7-s − 1.68i·11-s + 2.64i·13-s − 5.53i·17-s + 4.06·19-s + 8.17i·23-s − 25-s − 7.02·29-s + 5.21·31-s + (2.62 − 0.321i)35-s + 4.93·37-s + 1.96i·41-s − 0.922i·43-s + 9.67·47-s + ⋯ |
| L(s) = 1 | + 0.447i·5-s + (−0.121 − 0.992i)7-s − 0.508i·11-s + 0.732i·13-s − 1.34i·17-s + 0.933·19-s + 1.70i·23-s − 0.200·25-s − 1.30·29-s + 0.937·31-s + (0.443 − 0.0542i)35-s + 0.812·37-s + 0.307i·41-s − 0.140i·43-s + 1.41·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.601 + 0.798i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.601 + 0.798i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.745150612\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.745150612\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - iT \) |
| 7 | \( 1 + (0.321 + 2.62i)T \) |
| good | 11 | \( 1 + 1.68iT - 11T^{2} \) |
| 13 | \( 1 - 2.64iT - 13T^{2} \) |
| 17 | \( 1 + 5.53iT - 17T^{2} \) |
| 19 | \( 1 - 4.06T + 19T^{2} \) |
| 23 | \( 1 - 8.17iT - 23T^{2} \) |
| 29 | \( 1 + 7.02T + 29T^{2} \) |
| 31 | \( 1 - 5.21T + 31T^{2} \) |
| 37 | \( 1 - 4.93T + 37T^{2} \) |
| 41 | \( 1 - 1.96iT - 41T^{2} \) |
| 43 | \( 1 + 0.922iT - 43T^{2} \) |
| 47 | \( 1 - 9.67T + 47T^{2} \) |
| 53 | \( 1 - 0.279T + 53T^{2} \) |
| 59 | \( 1 - 3.49T + 59T^{2} \) |
| 61 | \( 1 + 14.1iT - 61T^{2} \) |
| 67 | \( 1 + 15.3iT - 67T^{2} \) |
| 71 | \( 1 - 10.6iT - 71T^{2} \) |
| 73 | \( 1 + 0.380iT - 73T^{2} \) |
| 79 | \( 1 + 0.0898iT - 79T^{2} \) |
| 83 | \( 1 + 14.4T + 83T^{2} \) |
| 89 | \( 1 + 5.54iT - 89T^{2} \) |
| 97 | \( 1 + 10.8iT - 97T^{2} \) |
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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
−7.85616634909666150004323403373, −7.38741718196874686497998688147, −6.88458156908166435494192645925, −5.97914132444802042327665149090, −5.23718629345250790581774078886, −4.33967633512499433081082430061, −3.54257420278046068805401600393, −2.90572661518539933429438774417, −1.67002642606982588092253116206, −0.57025161241795802035444920424,
0.954403421729108957816636707575, 2.13845974133767137745948150295, 2.85179575102047840904534153082, 3.93969303143789247049452226729, 4.65871274751918462818805492189, 5.64076498056703381036092749381, 5.91246748974973425224337027971, 6.90926155290392606762638433443, 7.75177476903604610120062653610, 8.423567852577036716295138743639
