| L(s) = 1 | − 4.56·2-s + 8.68·3-s + 12.8·4-s + 2.80·5-s − 39.6·6-s + 9.56·7-s − 21.9·8-s + 48.4·9-s − 12.8·10-s + 39.4·11-s + 111.·12-s − 43.6·14-s + 24.3·15-s − 2.42·16-s + 2.01·17-s − 220.·18-s − 60.1·19-s + 35.9·20-s + 83.0·21-s − 179.·22-s + 4.46·23-s − 190.·24-s − 117.·25-s + 186.·27-s + 122.·28-s + 140.·29-s − 111.·30-s + ⋯ |
| L(s) = 1 | − 1.61·2-s + 1.67·3-s + 1.60·4-s + 0.251·5-s − 2.69·6-s + 0.516·7-s − 0.969·8-s + 1.79·9-s − 0.405·10-s + 1.08·11-s + 2.67·12-s − 0.832·14-s + 0.419·15-s − 0.0378·16-s + 0.0287·17-s − 2.89·18-s − 0.726·19-s + 0.402·20-s + 0.862·21-s − 1.74·22-s + 0.0405·23-s − 1.61·24-s − 0.936·25-s + 1.32·27-s + 0.826·28-s + 0.900·29-s − 0.676·30-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
| \(L(2)\) |
\(\approx\) |
\(1.613881841\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.613881841\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 4.56T + 8T^{2} \) |
| 3 | \( 1 - 8.68T + 27T^{2} \) |
| 5 | \( 1 - 2.80T + 125T^{2} \) |
| 7 | \( 1 - 9.56T + 343T^{2} \) |
| 11 | \( 1 - 39.4T + 1.33e3T^{2} \) |
| 17 | \( 1 - 2.01T + 4.91e3T^{2} \) |
| 19 | \( 1 + 60.1T + 6.85e3T^{2} \) |
| 23 | \( 1 - 4.46T + 1.21e4T^{2} \) |
| 29 | \( 1 - 140.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 136.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 185.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 310.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 427.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 258.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 612.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 517.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 161.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 49.8T + 3.00e5T^{2} \) |
| 71 | \( 1 - 279.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 467.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 37.5T + 4.93e5T^{2} \) |
| 83 | \( 1 + 76.1T + 5.71e5T^{2} \) |
| 89 | \( 1 - 202.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.17e3T + 9.12e5T^{2} \) |
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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
−12.14449838035217235523903716250, −10.87666166523187653055936793541, −9.820472166915262012921323174683, −9.143952632366003963739965376817, −8.407289818671291103724780546323, −7.68766078346877634785645292826, −6.56210639532471586053115423337, −4.14831997915408472709190295504, −2.48295134284218711700162652248, −1.38775769839759439541310908148,
1.38775769839759439541310908148, 2.48295134284218711700162652248, 4.14831997915408472709190295504, 6.56210639532471586053115423337, 7.68766078346877634785645292826, 8.407289818671291103724780546323, 9.143952632366003963739965376817, 9.820472166915262012921323174683, 10.87666166523187653055936793541, 12.14449838035217235523903716250
