| L(s) = 1 | + 44·3-s − 1.22e3·7-s − 251·9-s + 3.16e3·11-s − 6.11e3·13-s + 1.62e4·17-s + 5.47e3·19-s − 5.38e4·21-s + 1.57e3·23-s − 1.07e5·27-s + 1.22e5·29-s − 2.51e5·31-s + 1.39e5·33-s + 5.23e4·37-s − 2.69e5·39-s − 3.19e5·41-s + 7.10e5·43-s + 2.84e5·47-s + 6.74e5·49-s + 7.15e5·51-s − 2.96e5·53-s + 2.40e5·57-s + 8.97e5·59-s − 8.84e5·61-s + 3.07e5·63-s + 4.65e6·67-s + 6.93e4·69-s + ⋯ |
| L(s) = 1 | + 0.940·3-s − 1.34·7-s − 0.114·9-s + 0.716·11-s − 0.772·13-s + 0.803·17-s + 0.183·19-s − 1.26·21-s + 0.0270·23-s − 1.04·27-s + 0.935·29-s − 1.51·31-s + 0.674·33-s + 0.169·37-s − 0.726·39-s − 0.723·41-s + 1.36·43-s + 0.399·47-s + 0.819·49-s + 0.755·51-s − 0.273·53-s + 0.172·57-s + 0.568·59-s − 0.499·61-s + 0.154·63-s + 1.89·67-s + 0.0254·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.268745935\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.268745935\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 44 T + p^{7} T^{2} \) |
| 7 | \( 1 + 1224 T + p^{7} T^{2} \) |
| 11 | \( 1 - 3164 T + p^{7} T^{2} \) |
| 13 | \( 1 + 6118 T + p^{7} T^{2} \) |
| 17 | \( 1 - 16270 T + p^{7} T^{2} \) |
| 19 | \( 1 - 5476 T + p^{7} T^{2} \) |
| 23 | \( 1 - 1576 T + p^{7} T^{2} \) |
| 29 | \( 1 - 122838 T + p^{7} T^{2} \) |
| 31 | \( 1 + 251360 T + p^{7} T^{2} \) |
| 37 | \( 1 - 52338 T + p^{7} T^{2} \) |
| 41 | \( 1 + 319398 T + p^{7} T^{2} \) |
| 43 | \( 1 - 710788 T + p^{7} T^{2} \) |
| 47 | \( 1 - 284112 T + p^{7} T^{2} \) |
| 53 | \( 1 + 296062 T + p^{7} T^{2} \) |
| 59 | \( 1 - 897548 T + p^{7} T^{2} \) |
| 61 | \( 1 + 884810 T + p^{7} T^{2} \) |
| 67 | \( 1 - 4659692 T + p^{7} T^{2} \) |
| 71 | \( 1 - 2710792 T + p^{7} T^{2} \) |
| 73 | \( 1 - 5670854 T + p^{7} T^{2} \) |
| 79 | \( 1 - 5124176 T + p^{7} T^{2} \) |
| 83 | \( 1 + 1563556 T + p^{7} T^{2} \) |
| 89 | \( 1 - 11605674 T + p^{7} T^{2} \) |
| 97 | \( 1 + 10931618 T + p^{7} T^{2} \) |
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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
−9.633188788548933368025901494049, −9.425566346211939930311139296730, −8.339825571666811184245178678640, −7.35147747238879489019783346162, −6.45528686364240946033345828722, −5.37161197664196074238405066595, −3.84318565936206031735157823709, −3.16129072067612101295088744418, −2.18804796101751560466542445313, −0.63916839610666779681167241819,
0.63916839610666779681167241819, 2.18804796101751560466542445313, 3.16129072067612101295088744418, 3.84318565936206031735157823709, 5.37161197664196074238405066595, 6.45528686364240946033345828722, 7.35147747238879489019783346162, 8.339825571666811184245178678640, 9.425566346211939930311139296730, 9.633188788548933368025901494049
