| L(s) = 1 | + 3·3-s − 5·5-s + 18.8·7-s + 9·9-s − 3.12·11-s − 76.2·13-s − 15·15-s + 66.6·17-s − 79.2·19-s + 56.4·21-s + 23·23-s + 25·25-s + 27·27-s − 203.·29-s + 217.·31-s − 9.37·33-s − 94.1·35-s + 74.3·37-s − 228.·39-s − 158.·41-s − 313.·43-s − 45·45-s − 599.·47-s + 11.3·49-s + 200.·51-s − 178.·53-s + 15.6·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1.01·7-s + 0.333·9-s − 0.0856·11-s − 1.62·13-s − 0.258·15-s + 0.951·17-s − 0.956·19-s + 0.586·21-s + 0.208·23-s + 0.200·25-s + 0.192·27-s − 1.30·29-s + 1.26·31-s − 0.0494·33-s − 0.454·35-s + 0.330·37-s − 0.939·39-s − 0.605·41-s − 1.11·43-s − 0.149·45-s − 1.85·47-s + 0.0330·49-s + 0.549·51-s − 0.463·53-s + 0.0383·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 3T \) |
| 5 | \( 1 + 5T \) |
| 23 | \( 1 - 23T \) |
| good | 7 | \( 1 - 18.8T + 343T^{2} \) |
| 11 | \( 1 + 3.12T + 1.33e3T^{2} \) |
| 13 | \( 1 + 76.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 66.6T + 4.91e3T^{2} \) |
| 19 | \( 1 + 79.2T + 6.85e3T^{2} \) |
| 29 | \( 1 + 203.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 217.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 74.3T + 5.06e4T^{2} \) |
| 41 | \( 1 + 158.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 313.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 599.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 178.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 203.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 290.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 250.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 616.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 445.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.15e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 372.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 304.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 411.T + 9.12e5T^{2} \) |
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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
−8.616071613117737171594430048197, −7.965680267748825956546449984224, −7.46947879294443984171645352362, −6.49825663918346621628170197790, −5.13460142230385892843409861159, −4.65861145825173274190661324046, −3.54856510142997798029708154288, −2.50188468471346285714917185979, −1.52459929886315129903608151432, 0,
1.52459929886315129903608151432, 2.50188468471346285714917185979, 3.54856510142997798029708154288, 4.65861145825173274190661324046, 5.13460142230385892843409861159, 6.49825663918346621628170197790, 7.46947879294443984171645352362, 7.965680267748825956546449984224, 8.616071613117737171594430048197
