| L(s) = 1 | − 0.189·2-s − 3.96·4-s + 9.91·5-s + 1.50·8-s + 9·9-s − 1.87·10-s + 14.1·11-s − 21.5·13-s + 15.5·16-s − 1.70·18-s − 21.3·19-s − 39.3·20-s − 2.67·22-s − 22.5·23-s + 73.3·25-s + 4.07·26-s − 58.7·31-s − 8.97·32-s − 35.6·36-s + 4.03·38-s + 14.9·40-s − 56.0·44-s + 89.2·45-s + 4.26·46-s + 49·49-s − 13.8·50-s + 85.4·52-s + ⋯ |
| L(s) = 1 | − 0.0945·2-s − 0.991·4-s + 1.98·5-s + 0.188·8-s + 9-s − 0.187·10-s + 1.28·11-s − 1.65·13-s + 0.973·16-s − 0.0945·18-s − 1.12·19-s − 1.96·20-s − 0.121·22-s − 0.981·23-s + 2.93·25-s + 0.156·26-s − 1.89·31-s − 0.280·32-s − 0.991·36-s + 0.106·38-s + 0.373·40-s − 1.27·44-s + 1.98·45-s + 0.0927·46-s + 0.999·49-s − 0.277·50-s + 1.64·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 79 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 79 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.349063244\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.349063244\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 79 | \( 1 + 79T \) |
| good | 2 | \( 1 + 0.189T + 4T^{2} \) |
| 3 | \( 1 - 9T^{2} \) |
| 5 | \( 1 - 9.91T + 25T^{2} \) |
| 7 | \( 1 - 49T^{2} \) |
| 11 | \( 1 - 14.1T + 121T^{2} \) |
| 13 | \( 1 + 21.5T + 169T^{2} \) |
| 17 | \( 1 - 289T^{2} \) |
| 19 | \( 1 + 21.3T + 361T^{2} \) |
| 23 | \( 1 + 22.5T + 529T^{2} \) |
| 29 | \( 1 - 841T^{2} \) |
| 31 | \( 1 + 58.7T + 961T^{2} \) |
| 37 | \( 1 - 1.36e3T^{2} \) |
| 41 | \( 1 - 1.68e3T^{2} \) |
| 43 | \( 1 - 1.84e3T^{2} \) |
| 47 | \( 1 - 2.20e3T^{2} \) |
| 53 | \( 1 - 2.80e3T^{2} \) |
| 59 | \( 1 - 3.48e3T^{2} \) |
| 61 | \( 1 - 3.72e3T^{2} \) |
| 67 | \( 1 - 55.5T + 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 + 118.T + 5.32e3T^{2} \) |
| 83 | \( 1 + 150T + 6.88e3T^{2} \) |
| 89 | \( 1 - 21.7T + 7.92e3T^{2} \) |
| 97 | \( 1 + 34.4T + 9.40e3T^{2} \) |
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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
−14.23752368082430122464574023142, −13.11127008332867942656564082207, −12.44582490601185352483617916591, −10.32470412519287995243817319707, −9.680816789947002012065122459256, −8.960861166871691111639307568788, −7.01961450428914279386780400658, −5.68982451835929812940080413435, −4.39018490791985509970205361613, −1.83926013976285273631992145864,
1.83926013976285273631992145864, 4.39018490791985509970205361613, 5.68982451835929812940080413435, 7.01961450428914279386780400658, 8.960861166871691111639307568788, 9.680816789947002012065122459256, 10.32470412519287995243817319707, 12.44582490601185352483617916591, 13.11127008332867942656564082207, 14.23752368082430122464574023142
