| L(s) = 1 | − 1.93·2-s + 3·3-s − 4.25·4-s + 7.94·5-s − 5.80·6-s − 25.7·7-s + 23.7·8-s + 9·9-s − 15.3·10-s + 11·11-s − 12.7·12-s + 82.1·13-s + 49.8·14-s + 23.8·15-s − 11.7·16-s − 9.84·17-s − 17.4·18-s − 24.3·19-s − 33.8·20-s − 77.3·21-s − 21.2·22-s − 99.0·23-s + 71.1·24-s − 61.8·25-s − 158.·26-s + 27·27-s + 109.·28-s + ⋯ |
| L(s) = 1 | − 0.683·2-s + 0.577·3-s − 0.532·4-s + 0.710·5-s − 0.394·6-s − 1.39·7-s + 1.04·8-s + 0.333·9-s − 0.486·10-s + 0.301·11-s − 0.307·12-s + 1.75·13-s + 0.952·14-s + 0.410·15-s − 0.183·16-s − 0.140·17-s − 0.227·18-s − 0.293·19-s − 0.378·20-s − 0.804·21-s − 0.206·22-s − 0.898·23-s + 0.604·24-s − 0.494·25-s − 1.19·26-s + 0.192·27-s + 0.741·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - 3T \) |
| 11 | \( 1 - 11T \) |
| 61 | \( 1 - 61T \) |
| good | 2 | \( 1 + 1.93T + 8T^{2} \) |
| 5 | \( 1 - 7.94T + 125T^{2} \) |
| 7 | \( 1 + 25.7T + 343T^{2} \) |
| 13 | \( 1 - 82.1T + 2.19e3T^{2} \) |
| 17 | \( 1 + 9.84T + 4.91e3T^{2} \) |
| 19 | \( 1 + 24.3T + 6.85e3T^{2} \) |
| 23 | \( 1 + 99.0T + 1.21e4T^{2} \) |
| 29 | \( 1 - 11.2T + 2.43e4T^{2} \) |
| 31 | \( 1 + 52.7T + 2.97e4T^{2} \) |
| 37 | \( 1 + 171.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 15.3T + 6.89e4T^{2} \) |
| 43 | \( 1 + 511.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 68.5T + 1.03e5T^{2} \) |
| 53 | \( 1 - 255.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 866.T + 2.05e5T^{2} \) |
| 67 | \( 1 + 246.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 851.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 722.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.27e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 585.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 631.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 710.T + 9.12e5T^{2} \) |
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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
−8.671715908438971007832164987544, −7.919538769091023959474425125973, −6.77183420637161905080882336901, −6.23820675800519131809513789025, −5.30646147491952288522666105186, −3.89460113019564349579361848778, −3.57594700957216746471180493016, −2.17388787256891786237316582785, −1.20600485751836057743549078406, 0,
1.20600485751836057743549078406, 2.17388787256891786237316582785, 3.57594700957216746471180493016, 3.89460113019564349579361848778, 5.30646147491952288522666105186, 6.23820675800519131809513789025, 6.77183420637161905080882336901, 7.919538769091023959474425125973, 8.671715908438971007832164987544
