| L(s) = 1 | + i·3-s − 1.33·7-s − 9-s + 2.94i·11-s + 2.04i·13-s + 3.61·17-s + 5.35i·19-s − 1.33i·21-s − 8.59·23-s − i·27-s − 5.26i·29-s + 2.08·31-s − 2.94·33-s − 6.55i·37-s − 2.04·39-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s − 0.504·7-s − 0.333·9-s + 0.887i·11-s + 0.566i·13-s + 0.876·17-s + 1.22i·19-s − 0.291i·21-s − 1.79·23-s − 0.192i·27-s − 0.977i·29-s + 0.373·31-s − 0.512·33-s − 1.07i·37-s − 0.326·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0418i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0418i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6368454517\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6368454517\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 1.33T + 7T^{2} \) |
| 11 | \( 1 - 2.94iT - 11T^{2} \) |
| 13 | \( 1 - 2.04iT - 13T^{2} \) |
| 17 | \( 1 - 3.61T + 17T^{2} \) |
| 19 | \( 1 - 5.35iT - 19T^{2} \) |
| 23 | \( 1 + 8.59T + 23T^{2} \) |
| 29 | \( 1 + 5.26iT - 29T^{2} \) |
| 31 | \( 1 - 2.08T + 31T^{2} \) |
| 37 | \( 1 + 6.55iT - 37T^{2} \) |
| 41 | \( 1 - 7.02T + 41T^{2} \) |
| 43 | \( 1 + 8.50iT - 43T^{2} \) |
| 47 | \( 1 + 9.97T + 47T^{2} \) |
| 53 | \( 1 - 6.12iT - 53T^{2} \) |
| 59 | \( 1 - 4.75iT - 59T^{2} \) |
| 61 | \( 1 - 8.51iT - 61T^{2} \) |
| 67 | \( 1 - 10.6iT - 67T^{2} \) |
| 71 | \( 1 - 2.62T + 71T^{2} \) |
| 73 | \( 1 + 15.3T + 73T^{2} \) |
| 79 | \( 1 + 10.4T + 79T^{2} \) |
| 83 | \( 1 - 1.52iT - 83T^{2} \) |
| 89 | \( 1 + 12.7T + 89T^{2} \) |
| 97 | \( 1 + 13.4T + 97T^{2} \) |
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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
−9.690358237597046897901099336773, −8.601276010998983167578526455683, −7.84564138832840941209173566209, −7.11476002884906384224583157644, −6.02826962205614890794162570497, −5.60913809919914896972506584472, −4.26001111045528283762388297972, −3.96859098492565967753482991710, −2.74271045399885477857810799138, −1.66782256482863956642016281477,
0.21102181212498836589006028465, 1.45517136832896105509126840696, 2.84143510674489984937913119799, 3.37399889332879633113902077523, 4.64446046095387487527167171961, 5.60863736743129249622165106332, 6.28124788760339544858488226438, 6.93728014399755242518936560510, 8.084986465413287726378927985037, 8.210311191222952185436009410343
