| L(s) = 1 | − 1.73·2-s − 0.732·3-s + 0.999·4-s − 1.73·5-s + 1.26·6-s − 4.73·7-s + 1.73·8-s − 2.46·9-s + 2.99·10-s − 1.26·11-s − 0.732·12-s + 13-s + 8.19·14-s + 1.26·15-s − 5·16-s − 3.46·17-s + 4.26·18-s + 4.73·19-s − 1.73·20-s + 3.46·21-s + 2.19·22-s − 1.26·24-s − 2.00·25-s − 1.73·26-s + 4·27-s − 4.73·28-s + 6.46·29-s + ⋯ |
| L(s) = 1 | − 1.22·2-s − 0.422·3-s + 0.499·4-s − 0.774·5-s + 0.517·6-s − 1.78·7-s + 0.612·8-s − 0.821·9-s + 0.948·10-s − 0.382·11-s − 0.211·12-s + 0.277·13-s + 2.19·14-s + 0.327·15-s − 1.25·16-s − 0.840·17-s + 1.00·18-s + 1.08·19-s − 0.387·20-s + 0.755·21-s + 0.468·22-s − 0.258·24-s − 0.400·25-s − 0.339·26-s + 0.769·27-s − 0.894·28-s + 1.20·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2297791709\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2297791709\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 \) |
| good | 2 | \( 1 + 1.73T + 2T^{2} \) |
| 3 | \( 1 + 0.732T + 3T^{2} \) |
| 5 | \( 1 + 1.73T + 5T^{2} \) |
| 7 | \( 1 + 4.73T + 7T^{2} \) |
| 11 | \( 1 + 1.26T + 11T^{2} \) |
| 13 | \( 1 - T + 13T^{2} \) |
| 17 | \( 1 + 3.46T + 17T^{2} \) |
| 19 | \( 1 - 4.73T + 19T^{2} \) |
| 29 | \( 1 - 6.46T + 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 + 6T + 37T^{2} \) |
| 41 | \( 1 - 0.464T + 41T^{2} \) |
| 43 | \( 1 + 0.928T + 43T^{2} \) |
| 47 | \( 1 - 7.26T + 47T^{2} \) |
| 53 | \( 1 - 7.73T + 53T^{2} \) |
| 59 | \( 1 - 8.19T + 59T^{2} \) |
| 61 | \( 1 + 0.803T + 61T^{2} \) |
| 67 | \( 1 - 2.53T + 67T^{2} \) |
| 71 | \( 1 - 10.7T + 71T^{2} \) |
| 73 | \( 1 + 15.3T + 73T^{2} \) |
| 79 | \( 1 - 2.53T + 79T^{2} \) |
| 83 | \( 1 + 6T + 83T^{2} \) |
| 89 | \( 1 + 4.26T + 89T^{2} \) |
| 97 | \( 1 + 1.73T + 97T^{2} \) |
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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
−10.61857596892217995123451019661, −9.931006341886570346105141172705, −9.048785468080599451349915960848, −8.400123451219807793483977029962, −7.30934167817666941540347125278, −6.57711484959963290752695123411, −5.43607354979392034959092506579, −3.94979673925827121955637759804, −2.77822697474233105785849701956, −0.49920237583885349159921848571,
0.49920237583885349159921848571, 2.77822697474233105785849701956, 3.94979673925827121955637759804, 5.43607354979392034959092506579, 6.57711484959963290752695123411, 7.30934167817666941540347125278, 8.400123451219807793483977029962, 9.048785468080599451349915960848, 9.931006341886570346105141172705, 10.61857596892217995123451019661
