| L(s) = 1 | + 4.06·2-s − 8.01·3-s + 8.48·4-s − 32.5·6-s + 18.3·7-s + 1.97·8-s + 37.2·9-s − 21.1·11-s − 68.0·12-s − 13·13-s + 74.5·14-s − 59.8·16-s + 1.67·17-s + 151.·18-s − 129.·19-s − 147.·21-s − 86.0·22-s + 8.41·23-s − 15.8·24-s − 52.7·26-s − 81.7·27-s + 155.·28-s − 138.·29-s − 203.·31-s − 258.·32-s + 169.·33-s + 6.81·34-s + ⋯ |
| L(s) = 1 | + 1.43·2-s − 1.54·3-s + 1.06·4-s − 2.21·6-s + 0.991·7-s + 0.0875·8-s + 1.37·9-s − 0.580·11-s − 1.63·12-s − 0.277·13-s + 1.42·14-s − 0.935·16-s + 0.0239·17-s + 1.97·18-s − 1.56·19-s − 1.52·21-s − 0.833·22-s + 0.0762·23-s − 0.134·24-s − 0.398·26-s − 0.582·27-s + 1.05·28-s − 0.885·29-s − 1.18·31-s − 1.43·32-s + 0.895·33-s + 0.0343·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 13 | \( 1 + 13T \) |
| good | 2 | \( 1 - 4.06T + 8T^{2} \) |
| 3 | \( 1 + 8.01T + 27T^{2} \) |
| 7 | \( 1 - 18.3T + 343T^{2} \) |
| 11 | \( 1 + 21.1T + 1.33e3T^{2} \) |
| 17 | \( 1 - 1.67T + 4.91e3T^{2} \) |
| 19 | \( 1 + 129.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 8.41T + 1.21e4T^{2} \) |
| 29 | \( 1 + 138.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 203.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 376.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 62.4T + 6.89e4T^{2} \) |
| 43 | \( 1 + 355.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 600.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 192.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 661.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 770.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 821.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 866.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 227.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 3.26T + 4.93e5T^{2} \) |
| 83 | \( 1 - 316.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.13e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 658.T + 9.12e5T^{2} \) |
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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
−11.03919959852629740985523475111, −10.43157907737840636630389710310, −8.770973027563245724278012658153, −7.33904856017782299591957543020, −6.35372267632795843402176668502, −5.40125690212375621146169939003, −4.93390160190217179013931131770, −3.89884063465540720468199989247, −2.04198384812628974837077522805, 0,
2.04198384812628974837077522805, 3.89884063465540720468199989247, 4.93390160190217179013931131770, 5.40125690212375621146169939003, 6.35372267632795843402176668502, 7.33904856017782299591957543020, 8.770973027563245724278012658153, 10.43157907737840636630389710310, 11.03919959852629740985523475111
