| L(s) = 1 | + 5.49·2-s + 6.46·3-s + 22.1·4-s + 35.5·6-s − 20.3·7-s + 77.9·8-s + 14.7·9-s + 52.0·11-s + 143.·12-s − 7.04·13-s − 111.·14-s + 251.·16-s − 28.7·17-s + 81.1·18-s + 76.4·19-s − 131.·21-s + 286.·22-s − 59.7·23-s + 504.·24-s − 38.7·26-s − 79.0·27-s − 451.·28-s − 29·29-s − 3.25·31-s + 755.·32-s + 336.·33-s − 158.·34-s + ⋯ |
| L(s) = 1 | + 1.94·2-s + 1.24·3-s + 2.77·4-s + 2.41·6-s − 1.09·7-s + 3.44·8-s + 0.547·9-s + 1.42·11-s + 3.45·12-s − 0.150·13-s − 2.13·14-s + 3.92·16-s − 0.410·17-s + 1.06·18-s + 0.922·19-s − 1.36·21-s + 2.77·22-s − 0.541·23-s + 4.28·24-s − 0.292·26-s − 0.563·27-s − 3.04·28-s − 0.185·29-s − 0.0188·31-s + 4.17·32-s + 1.77·33-s − 0.797·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(10.56929998\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.56929998\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 29 | \( 1 + 29T \) |
| good | 2 | \( 1 - 5.49T + 8T^{2} \) |
| 3 | \( 1 - 6.46T + 27T^{2} \) |
| 7 | \( 1 + 20.3T + 343T^{2} \) |
| 11 | \( 1 - 52.0T + 1.33e3T^{2} \) |
| 13 | \( 1 + 7.04T + 2.19e3T^{2} \) |
| 17 | \( 1 + 28.7T + 4.91e3T^{2} \) |
| 19 | \( 1 - 76.4T + 6.85e3T^{2} \) |
| 23 | \( 1 + 59.7T + 1.21e4T^{2} \) |
| 31 | \( 1 + 3.25T + 2.97e4T^{2} \) |
| 37 | \( 1 + 150.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 92.3T + 6.89e4T^{2} \) |
| 43 | \( 1 - 100.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 324.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 374.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 489.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 221.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 427.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 898.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.08e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 798.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 436.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 456.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 803.T + 9.12e5T^{2} \) |
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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
−9.980989539188926390225910867365, −9.251369119457969874130580734793, −8.037795491401575753241888660972, −6.97408539682641261506410345185, −6.45324759762860074473445547589, −5.41244737911965527275514513566, −4.11973383927290425036346055603, −3.52564540690672522574694356783, −2.81497761464972338251297771114, −1.69999340718251665106249863682,
1.69999340718251665106249863682, 2.81497761464972338251297771114, 3.52564540690672522574694356783, 4.11973383927290425036346055603, 5.41244737911965527275514513566, 6.45324759762860074473445547589, 6.97408539682641261506410345185, 8.037795491401575753241888660972, 9.251369119457969874130580734793, 9.980989539188926390225910867365
