| L(s) = 1 | + 2-s + 2.65·3-s + 4-s + 5-s + 2.65·6-s + 8-s + 4.04·9-s + 10-s − 2.65·11-s + 2.65·12-s + 2.75·13-s + 2.65·15-s + 16-s + 17-s + 4.04·18-s − 2.40·19-s + 20-s − 2.65·22-s + 0.278·23-s + 2.65·24-s + 25-s + 2.75·26-s + 2.78·27-s + 4.76·29-s + 2.65·30-s + 0.128·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.53·3-s + 0.5·4-s + 0.447·5-s + 1.08·6-s + 0.353·8-s + 1.34·9-s + 0.316·10-s − 0.799·11-s + 0.766·12-s + 0.763·13-s + 0.685·15-s + 0.250·16-s + 0.242·17-s + 0.954·18-s − 0.551·19-s + 0.223·20-s − 0.565·22-s + 0.0581·23-s + 0.541·24-s + 0.200·25-s + 0.539·26-s + 0.535·27-s + 0.884·29-s + 0.484·30-s + 0.0230·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.797595395\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.797595395\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 - 2.65T + 3T^{2} \) |
| 11 | \( 1 + 2.65T + 11T^{2} \) |
| 13 | \( 1 - 2.75T + 13T^{2} \) |
| 19 | \( 1 + 2.40T + 19T^{2} \) |
| 23 | \( 1 - 0.278T + 23T^{2} \) |
| 29 | \( 1 - 4.76T + 29T^{2} \) |
| 31 | \( 1 - 0.128T + 31T^{2} \) |
| 37 | \( 1 - 4.04T + 37T^{2} \) |
| 41 | \( 1 - 6.52T + 41T^{2} \) |
| 43 | \( 1 + 2.39T + 43T^{2} \) |
| 47 | \( 1 + 6.58T + 47T^{2} \) |
| 53 | \( 1 - 9.52T + 53T^{2} \) |
| 59 | \( 1 - 8.77T + 59T^{2} \) |
| 61 | \( 1 - 4.25T + 61T^{2} \) |
| 67 | \( 1 - 15.6T + 67T^{2} \) |
| 71 | \( 1 + 3.67T + 71T^{2} \) |
| 73 | \( 1 - 10.9T + 73T^{2} \) |
| 79 | \( 1 - 3.34T + 79T^{2} \) |
| 83 | \( 1 - 2.20T + 83T^{2} \) |
| 89 | \( 1 + 8.32T + 89T^{2} \) |
| 97 | \( 1 - 9.30T + 97T^{2} \) |
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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
−8.059077522576073515516925225551, −7.10028979446900276153406764550, −6.47297511954370694299799548367, −5.65840452062536507392916990236, −4.93074353890126790585764858138, −4.03276059083171099726552333434, −3.49406214162718500514754951563, −2.58864909714357481852868882326, −2.26387705693069143729583165993, −1.12634575808581476626584991234,
1.12634575808581476626584991234, 2.26387705693069143729583165993, 2.58864909714357481852868882326, 3.49406214162718500514754951563, 4.03276059083171099726552333434, 4.93074353890126790585764858138, 5.65840452062536507392916990236, 6.47297511954370694299799548367, 7.10028979446900276153406764550, 8.059077522576073515516925225551
