| L(s) = 1 | − 1.41i·5-s − 4i·7-s − 5.65·11-s − 4·13-s + 4.24i·17-s − 5.65·23-s + 2.99·25-s + 1.41i·29-s − 4i·31-s − 5.65·35-s + 6·37-s − 9.89i·41-s − 8i·43-s − 5.65·47-s − 9·49-s + ⋯ |
| L(s) = 1 | − 0.632i·5-s − 1.51i·7-s − 1.70·11-s − 1.10·13-s + 1.02i·17-s − 1.17·23-s + 0.599·25-s + 0.262i·29-s − 0.718i·31-s − 0.956·35-s + 0.986·37-s − 1.54i·41-s − 1.21i·43-s − 0.825·47-s − 1.28·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 + 0.577i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.229202 - 0.721131i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.229202 - 0.721131i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 1.41iT - 5T^{2} \) |
| 7 | \( 1 + 4iT - 7T^{2} \) |
| 11 | \( 1 + 5.65T + 11T^{2} \) |
| 13 | \( 1 + 4T + 13T^{2} \) |
| 17 | \( 1 - 4.24iT - 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + 5.65T + 23T^{2} \) |
| 29 | \( 1 - 1.41iT - 29T^{2} \) |
| 31 | \( 1 + 4iT - 31T^{2} \) |
| 37 | \( 1 - 6T + 37T^{2} \) |
| 41 | \( 1 + 9.89iT - 41T^{2} \) |
| 43 | \( 1 + 8iT - 43T^{2} \) |
| 47 | \( 1 + 5.65T + 47T^{2} \) |
| 53 | \( 1 + 4.24iT - 53T^{2} \) |
| 59 | \( 1 - 11.3T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + 8iT - 67T^{2} \) |
| 71 | \( 1 - 5.65T + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 - 4iT - 79T^{2} \) |
| 83 | \( 1 - 5.65T + 83T^{2} \) |
| 89 | \( 1 + 4.24iT - 89T^{2} \) |
| 97 | \( 1 + 8T + 97T^{2} \) |
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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
−10.32360937725205930273853100162, −9.776610372636196852004421041476, −8.380263130328563341616889716055, −7.75379788807592481996354273366, −6.96603639246896978904741026584, −5.59283793293401047972675320564, −4.70885502241919532450136180327, −3.76218282389824909556386566466, −2.20165118443037188677233009361, −0.39560713202295222952378087818,
2.47865409445110051051012942284, 2.83355167419338710386771526045, 4.80126549567677412640917389036, 5.47682241251627425783245967948, 6.52109639569291087815479229536, 7.64447280433074367089497536340, 8.301592137560275702336947977761, 9.509092865350021842439451809237, 10.07033225462411733337044380435, 11.13485778523077898451014663513
