| L(s) = 1 | + 20.2·5-s + 14.9·7-s + 55.0·11-s − 13·13-s − 83.5·17-s − 64.3·19-s − 21.1·23-s + 286.·25-s + 269.·29-s − 159.·31-s + 303.·35-s + 156.·37-s + 472.·41-s − 364.·43-s + 8.13·47-s − 119.·49-s + 640.·53-s + 1.11e3·55-s + 442.·59-s + 271.·61-s − 263.·65-s + 714.·67-s − 1.12e3·71-s + 425.·73-s + 822.·77-s − 12.3·79-s − 475.·83-s + ⋯ |
| L(s) = 1 | + 1.81·5-s + 0.806·7-s + 1.50·11-s − 0.277·13-s − 1.19·17-s − 0.776·19-s − 0.191·23-s + 2.29·25-s + 1.72·29-s − 0.922·31-s + 1.46·35-s + 0.695·37-s + 1.79·41-s − 1.29·43-s + 0.0252·47-s − 0.349·49-s + 1.65·53-s + 2.73·55-s + 0.977·59-s + 0.570·61-s − 0.503·65-s + 1.30·67-s − 1.88·71-s + 0.682·73-s + 1.21·77-s − 0.0175·79-s − 0.628·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.101383872\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.101383872\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 + 13T \) |
| good | 5 | \( 1 - 20.2T + 125T^{2} \) |
| 7 | \( 1 - 14.9T + 343T^{2} \) |
| 11 | \( 1 - 55.0T + 1.33e3T^{2} \) |
| 17 | \( 1 + 83.5T + 4.91e3T^{2} \) |
| 19 | \( 1 + 64.3T + 6.85e3T^{2} \) |
| 23 | \( 1 + 21.1T + 1.21e4T^{2} \) |
| 29 | \( 1 - 269.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 159.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 156.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 472.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 364.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 8.13T + 1.03e5T^{2} \) |
| 53 | \( 1 - 640.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 442.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 271.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 714.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.12e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 425.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 12.3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 475.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 302.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.24e3T + 9.12e5T^{2} \) |
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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
−8.915777952794354009612025873827, −8.395696442935371946234438764199, −6.98672060998667312800834643042, −6.47556036209014060901459343212, −5.76663480079656310993450931466, −4.82575651737153552524485163280, −4.09268290159145792317328070068, −2.54591724813866756045013647559, −1.89696638219912390203250864842, −1.01471752348186714701707339569,
1.01471752348186714701707339569, 1.89696638219912390203250864842, 2.54591724813866756045013647559, 4.09268290159145792317328070068, 4.82575651737153552524485163280, 5.76663480079656310993450931466, 6.47556036209014060901459343212, 6.98672060998667312800834643042, 8.395696442935371946234438764199, 8.915777952794354009612025873827
