| L(s) = 1 | + 2-s + 3.33·3-s + 4-s + 5-s + 3.33·6-s − 1.43·7-s + 8-s + 8.10·9-s + 10-s − 4.03·11-s + 3.33·12-s − 1.43·14-s + 3.33·15-s + 16-s + 2.30·17-s + 8.10·18-s + 0.399·19-s + 20-s − 4.79·21-s − 4.03·22-s + 1.56·23-s + 3.33·24-s + 25-s + 17.0·27-s − 1.43·28-s − 7.86·29-s + 3.33·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.92·3-s + 0.5·4-s + 0.447·5-s + 1.36·6-s − 0.544·7-s + 0.353·8-s + 2.70·9-s + 0.316·10-s − 1.21·11-s + 0.961·12-s − 0.384·14-s + 0.860·15-s + 0.250·16-s + 0.559·17-s + 1.91·18-s + 0.0917·19-s + 0.223·20-s − 1.04·21-s − 0.861·22-s + 0.325·23-s + 0.680·24-s + 0.200·25-s + 3.27·27-s − 0.272·28-s − 1.46·29-s + 0.608·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.254860529\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.254860529\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - 3.33T + 3T^{2} \) |
| 7 | \( 1 + 1.43T + 7T^{2} \) |
| 11 | \( 1 + 4.03T + 11T^{2} \) |
| 17 | \( 1 - 2.30T + 17T^{2} \) |
| 19 | \( 1 - 0.399T + 19T^{2} \) |
| 23 | \( 1 - 1.56T + 23T^{2} \) |
| 29 | \( 1 + 7.86T + 29T^{2} \) |
| 31 | \( 1 - 3.10T + 31T^{2} \) |
| 37 | \( 1 - 6.23T + 37T^{2} \) |
| 41 | \( 1 + 0.761T + 41T^{2} \) |
| 43 | \( 1 + 11.0T + 43T^{2} \) |
| 47 | \( 1 - 5.84T + 47T^{2} \) |
| 53 | \( 1 - 10.6T + 53T^{2} \) |
| 59 | \( 1 + 3.46T + 59T^{2} \) |
| 61 | \( 1 + 5.69T + 61T^{2} \) |
| 67 | \( 1 + 10.4T + 67T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 + 13.4T + 73T^{2} \) |
| 79 | \( 1 - 0.931T + 79T^{2} \) |
| 83 | \( 1 + 10.3T + 83T^{2} \) |
| 89 | \( 1 - 0.971T + 89T^{2} \) |
| 97 | \( 1 - 12.3T + 97T^{2} \) |
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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
−9.309960541434025569576247706427, −8.538344604815481892836774074819, −7.67656910136982302464249709199, −7.23129842282639285836983627627, −6.11453866138094220824895446254, −5.08384596847434891257571972897, −4.10019987996383497027185865356, −3.14647441450112652510867717922, −2.68427580814046590517766750158, −1.64999396455490597890823565744,
1.64999396455490597890823565744, 2.68427580814046590517766750158, 3.14647441450112652510867717922, 4.10019987996383497027185865356, 5.08384596847434891257571972897, 6.11453866138094220824895446254, 7.23129842282639285836983627627, 7.67656910136982302464249709199, 8.538344604815481892836774074819, 9.309960541434025569576247706427
