| L(s) = 1 | + (−4.99 − 0.311i)5-s − 4.65i·7-s − 12.0i·11-s − 8.46i·13-s + 1.25·17-s + 16.4·19-s − 2.38·23-s + (24.8 + 3.10i)25-s − 23.9i·29-s − 5.25·31-s + (−1.44 + 23.2i)35-s + 25.1i·37-s + 29.8i·41-s − 25.5i·43-s − 64.9·47-s + ⋯ |
| L(s) = 1 | + (−0.998 − 0.0622i)5-s − 0.665i·7-s − 1.09i·11-s − 0.651i·13-s + 0.0736·17-s + 0.867·19-s − 0.103·23-s + (0.992 + 0.124i)25-s − 0.827i·29-s − 0.169·31-s + (−0.0413 + 0.663i)35-s + 0.680i·37-s + 0.727i·41-s − 0.593i·43-s − 1.38·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0622i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.998 - 0.0622i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.5729721150\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5729721150\) |
| \(L(2)\) |
|
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 | \( 1 + (4.99 + 0.311i)T \) |
| good | 7 | \( 1 + 4.65iT - 49T^{2} \) |
| 11 | \( 1 + 12.0iT - 121T^{2} \) |
| 13 | \( 1 + 8.46iT - 169T^{2} \) |
| 17 | \( 1 - 1.25T + 289T^{2} \) |
| 19 | \( 1 - 16.4T + 361T^{2} \) |
| 23 | \( 1 + 2.38T + 529T^{2} \) |
| 29 | \( 1 + 23.9iT - 841T^{2} \) |
| 31 | \( 1 + 5.25T + 961T^{2} \) |
| 37 | \( 1 - 25.1iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 29.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 25.5iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 64.9T + 2.20e3T^{2} \) |
| 53 | \( 1 + 71.4T + 2.80e3T^{2} \) |
| 59 | \( 1 + 18.5iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 19.1T + 3.72e3T^{2} \) |
| 67 | \( 1 + 81.0iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 21.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 109. iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 134.T + 6.24e3T^{2} \) |
| 83 | \( 1 + 15.6T + 6.88e3T^{2} \) |
| 89 | \( 1 + 79.4iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 130. iT - 9.40e3T^{2} \) |
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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
−8.566909663175379025791524365532, −8.017752831025993281543321643142, −7.36865917979190112490602160270, −6.44195440093976785250233899758, −5.47556935769975859852984638174, −4.53632179592387648811513616132, −3.59012975493983806675161667308, −2.96844527063485614681914477756, −1.15676995086725380187558174880, −0.17496730127465045914190993389,
1.51259954090591246968757114511, 2.71332963337563208726267429661, 3.73783506083536981246671178300, 4.62556325960652652830401336206, 5.39000653904169495261737167750, 6.57160650369970258233695756675, 7.28054450202785280368634768557, 7.933638855290497519370839474464, 8.887876554683059831934616806398, 9.457648666702704531460573949178
