| L(s) = 1 | + 2.70·2-s − 3-s + 5.34·4-s − 2.70·6-s − 1.07·7-s + 9.04·8-s + 9-s + 11-s − 5.34·12-s + 4.34·13-s − 2.92·14-s + 13.8·16-s − 7.75·17-s + 2.70·18-s + 5.26·19-s + 1.07·21-s + 2.70·22-s + 2.15·23-s − 9.04·24-s + 11.7·26-s − 27-s − 5.75·28-s + 1.41·29-s − 4.68·31-s + 19.3·32-s − 33-s − 21.0·34-s + ⋯ |
| L(s) = 1 | + 1.91·2-s − 0.577·3-s + 2.67·4-s − 1.10·6-s − 0.407·7-s + 3.19·8-s + 0.333·9-s + 0.301·11-s − 1.54·12-s + 1.20·13-s − 0.780·14-s + 3.45·16-s − 1.88·17-s + 0.638·18-s + 1.20·19-s + 0.235·21-s + 0.577·22-s + 0.449·23-s − 1.84·24-s + 2.30·26-s − 0.192·27-s − 1.08·28-s + 0.263·29-s − 0.840·31-s + 3.42·32-s − 0.174·33-s − 3.60·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.256474993\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.256474993\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 - 2.70T + 2T^{2} \) |
| 7 | \( 1 + 1.07T + 7T^{2} \) |
| 13 | \( 1 - 4.34T + 13T^{2} \) |
| 17 | \( 1 + 7.75T + 17T^{2} \) |
| 19 | \( 1 - 5.26T + 19T^{2} \) |
| 23 | \( 1 - 2.15T + 23T^{2} \) |
| 29 | \( 1 - 1.41T + 29T^{2} \) |
| 31 | \( 1 + 4.68T + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 + 9.41T + 41T^{2} \) |
| 43 | \( 1 + 7.60T + 43T^{2} \) |
| 47 | \( 1 + 4.68T + 47T^{2} \) |
| 53 | \( 1 + 0.156T + 53T^{2} \) |
| 59 | \( 1 - 6.15T + 59T^{2} \) |
| 61 | \( 1 + 4.15T + 61T^{2} \) |
| 67 | \( 1 - 8.68T + 67T^{2} \) |
| 71 | \( 1 + 4.68T + 71T^{2} \) |
| 73 | \( 1 - 10.4T + 73T^{2} \) |
| 79 | \( 1 + 8.09T + 79T^{2} \) |
| 83 | \( 1 - 11.0T + 83T^{2} \) |
| 89 | \( 1 + 12.8T + 89T^{2} \) |
| 97 | \( 1 + 14.6T + 97T^{2} \) |
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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
−10.82709174380023395352038537089, −9.630798721978810368269117603607, −8.318272012970896134030915355471, −6.86355015065206450571995951020, −6.65996652849161641384712338128, −5.66318824732485087487866037554, −4.88109582633404357410849875738, −3.93947098832744351545324993545, −3.10743837981281914408068596125, −1.67763356661225371788350724459,
1.67763356661225371788350724459, 3.10743837981281914408068596125, 3.93947098832744351545324993545, 4.88109582633404357410849875738, 5.66318824732485087487866037554, 6.65996652849161641384712338128, 6.86355015065206450571995951020, 8.318272012970896134030915355471, 9.630798721978810368269117603607, 10.82709174380023395352038537089
