| L(s) = 1 | + 1.73·2-s + 2·3-s − 5·4-s + 1.73·5-s + 3.46·6-s − 13.8·7-s − 22.5·8-s − 23·9-s + 2.99·10-s − 13.8·11-s − 10·12-s − 23.9·14-s + 3.46·15-s + 1.00·16-s − 117·17-s − 39.8·18-s + 114.·19-s − 8.66·20-s − 27.7·21-s − 23.9·22-s + 78·23-s − 45.0·24-s − 122·25-s − 100·27-s + 69.2·28-s − 141·29-s + 5.99·30-s + ⋯ |
| L(s) = 1 | + 0.612·2-s + 0.384·3-s − 0.625·4-s + 0.154·5-s + 0.235·6-s − 0.748·7-s − 0.995·8-s − 0.851·9-s + 0.0948·10-s − 0.379·11-s − 0.240·12-s − 0.458·14-s + 0.0596·15-s + 0.0156·16-s − 1.66·17-s − 0.521·18-s + 1.38·19-s − 0.0968·20-s − 0.287·21-s − 0.232·22-s + 0.707·23-s − 0.383·24-s − 0.975·25-s − 0.712·27-s + 0.467·28-s − 0.902·29-s + 0.0365·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 13 | \( 1 \) |
| good | 2 | \( 1 - 1.73T + 8T^{2} \) |
| 3 | \( 1 - 2T + 27T^{2} \) |
| 5 | \( 1 - 1.73T + 125T^{2} \) |
| 7 | \( 1 + 13.8T + 343T^{2} \) |
| 11 | \( 1 + 13.8T + 1.33e3T^{2} \) |
| 17 | \( 1 + 117T + 4.91e3T^{2} \) |
| 19 | \( 1 - 114.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 78T + 1.21e4T^{2} \) |
| 29 | \( 1 + 141T + 2.43e4T^{2} \) |
| 31 | \( 1 + 155.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 143.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 271.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 104T + 7.95e4T^{2} \) |
| 47 | \( 1 - 301.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 93T + 1.48e5T^{2} \) |
| 59 | \( 1 + 284.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 145T + 2.26e5T^{2} \) |
| 67 | \( 1 + 786.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.05e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 458.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.27e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 789.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 976.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 200.T + 9.12e5T^{2} \) |
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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
−12.01499444193470018571559050193, −10.95433303549758176094023366031, −9.420453313996387944752541398273, −9.055370212405757343801346837655, −7.67492109819294819179643282623, −6.18599457548041629508382113193, −5.22518062639049430952981343378, −3.80813782988406630749550864098, −2.69016200993664497503055919264, 0,
2.69016200993664497503055919264, 3.80813782988406630749550864098, 5.22518062639049430952981343378, 6.18599457548041629508382113193, 7.67492109819294819179643282623, 9.055370212405757343801346837655, 9.420453313996387944752541398273, 10.95433303549758176094023366031, 12.01499444193470018571559050193
