| L(s) = 1 | + 1.71·3-s + 0.193·5-s − 0.0502·9-s + 4.62·11-s − 1.46·13-s + 0.332·15-s − 3.70·17-s − 4.29·19-s + 2.42·23-s − 4.96·25-s − 5.23·27-s − 5.58·29-s − 8.73·31-s + 7.93·33-s + 1.47·37-s − 2.51·39-s − 5.02·41-s − 10.3·43-s − 0.00971·45-s − 2.92·47-s − 6.35·51-s − 1.84·53-s + 0.894·55-s − 7.38·57-s − 1.25·59-s + 3.25·61-s − 0.283·65-s + ⋯ |
| L(s) = 1 | + 0.991·3-s + 0.0864·5-s − 0.0167·9-s + 1.39·11-s − 0.405·13-s + 0.0857·15-s − 0.897·17-s − 0.986·19-s + 0.505·23-s − 0.992·25-s − 1.00·27-s − 1.03·29-s − 1.56·31-s + 1.38·33-s + 0.241·37-s − 0.402·39-s − 0.784·41-s − 1.57·43-s − 0.00144·45-s − 0.426·47-s − 0.889·51-s − 0.253·53-s + 0.120·55-s − 0.977·57-s − 0.163·59-s + 0.416·61-s − 0.0351·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5488 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5488 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 1.71T + 3T^{2} \) |
| 5 | \( 1 - 0.193T + 5T^{2} \) |
| 11 | \( 1 - 4.62T + 11T^{2} \) |
| 13 | \( 1 + 1.46T + 13T^{2} \) |
| 17 | \( 1 + 3.70T + 17T^{2} \) |
| 19 | \( 1 + 4.29T + 19T^{2} \) |
| 23 | \( 1 - 2.42T + 23T^{2} \) |
| 29 | \( 1 + 5.58T + 29T^{2} \) |
| 31 | \( 1 + 8.73T + 31T^{2} \) |
| 37 | \( 1 - 1.47T + 37T^{2} \) |
| 41 | \( 1 + 5.02T + 41T^{2} \) |
| 43 | \( 1 + 10.3T + 43T^{2} \) |
| 47 | \( 1 + 2.92T + 47T^{2} \) |
| 53 | \( 1 + 1.84T + 53T^{2} \) |
| 59 | \( 1 + 1.25T + 59T^{2} \) |
| 61 | \( 1 - 3.25T + 61T^{2} \) |
| 67 | \( 1 - 10.9T + 67T^{2} \) |
| 71 | \( 1 + 4.57T + 71T^{2} \) |
| 73 | \( 1 + 1.74T + 73T^{2} \) |
| 79 | \( 1 + 2.06T + 79T^{2} \) |
| 83 | \( 1 - 9.68T + 83T^{2} \) |
| 89 | \( 1 - 16.8T + 89T^{2} \) |
| 97 | \( 1 - 13.4T + 97T^{2} \) |
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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
−7.945979894539439655229873243672, −7.08392082608214294854031550049, −6.52892602494584581412607972211, −5.69975023860737509487519709845, −4.74870239302011796063802894301, −3.83466475181870429282167359463, −3.42371642780509298367568835988, −2.20668314085145612223295440593, −1.74033255835820376543343780802, 0,
1.74033255835820376543343780802, 2.20668314085145612223295440593, 3.42371642780509298367568835988, 3.83466475181870429282167359463, 4.74870239302011796063802894301, 5.69975023860737509487519709845, 6.52892602494584581412607972211, 7.08392082608214294854031550049, 7.945979894539439655229873243672
