| L(s) = 1 | + 1.41·3-s − 5-s − 0.999·9-s + 1.41·11-s + 13-s − 1.41·15-s − 2·17-s − 1.41·19-s + 7.07·23-s + 25-s − 5.65·27-s + 8·29-s + 9.89·31-s + 2.00·33-s + 4·37-s + 1.41·39-s + 6·41-s + 1.41·43-s + 0.999·45-s − 7·49-s − 2.82·51-s + 6·53-s − 1.41·55-s − 2.00·57-s − 4.24·59-s + 4·61-s − 65-s + ⋯ |
| L(s) = 1 | + 0.816·3-s − 0.447·5-s − 0.333·9-s + 0.426·11-s + 0.277·13-s − 0.365·15-s − 0.485·17-s − 0.324·19-s + 1.47·23-s + 0.200·25-s − 1.08·27-s + 1.48·29-s + 1.77·31-s + 0.348·33-s + 0.657·37-s + 0.226·39-s + 0.937·41-s + 0.215·43-s + 0.149·45-s − 49-s − 0.396·51-s + 0.824·53-s − 0.190·55-s − 0.264·57-s − 0.552·59-s + 0.512·61-s − 0.124·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.167120601\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.167120601\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 - 1.41T + 3T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 - 1.41T + 11T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 + 1.41T + 19T^{2} \) |
| 23 | \( 1 - 7.07T + 23T^{2} \) |
| 29 | \( 1 - 8T + 29T^{2} \) |
| 31 | \( 1 - 9.89T + 31T^{2} \) |
| 37 | \( 1 - 4T + 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - 1.41T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 + 4.24T + 59T^{2} \) |
| 61 | \( 1 - 4T + 61T^{2} \) |
| 67 | \( 1 - 14.1T + 67T^{2} \) |
| 71 | \( 1 + 15.5T + 71T^{2} \) |
| 73 | \( 1 - 4T + 73T^{2} \) |
| 79 | \( 1 + 2.82T + 79T^{2} \) |
| 83 | \( 1 + 5.65T + 83T^{2} \) |
| 89 | \( 1 - 14T + 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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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
−8.898032394061960257620666360562, −8.456690946247999466342992528880, −7.75158284399719743795664268979, −6.79098212257702470924986475172, −6.13011603626239084509450323436, −4.90130892366939920906866174764, −4.15264857376176038887777634895, −3.13337481145512069473941726136, −2.48021950040149067650063118398, −0.962110253747872861061071414904,
0.962110253747872861061071414904, 2.48021950040149067650063118398, 3.13337481145512069473941726136, 4.15264857376176038887777634895, 4.90130892366939920906866174764, 6.13011603626239084509450323436, 6.79098212257702470924986475172, 7.75158284399719743795664268979, 8.456690946247999466342992528880, 8.898032394061960257620666360562
