| L(s) = 1 | + 2.21·2-s − 3.08·4-s + 22.0·5-s − 7·7-s − 24.5·8-s + 48.8·10-s − 11·11-s + 2.76·13-s − 15.5·14-s − 29.7·16-s + 50.6·17-s + 56.5·19-s − 68.0·20-s − 24.3·22-s + 172.·23-s + 360.·25-s + 6.13·26-s + 21.6·28-s − 282.·29-s + 237.·31-s + 130.·32-s + 112.·34-s − 154.·35-s + 207.·37-s + 125.·38-s − 541.·40-s − 386.·41-s + ⋯ |
| L(s) = 1 | + 0.783·2-s − 0.386·4-s + 1.97·5-s − 0.377·7-s − 1.08·8-s + 1.54·10-s − 0.301·11-s + 0.0590·13-s − 0.296·14-s − 0.464·16-s + 0.722·17-s + 0.682·19-s − 0.761·20-s − 0.236·22-s + 1.56·23-s + 2.88·25-s + 0.0462·26-s + 0.145·28-s − 1.80·29-s + 1.37·31-s + 0.721·32-s + 0.566·34-s − 0.745·35-s + 0.920·37-s + 0.534·38-s − 2.14·40-s − 1.47·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.654576454\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.654576454\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| 11 | \( 1 + 11T \) |
| good | 2 | \( 1 - 2.21T + 8T^{2} \) |
| 5 | \( 1 - 22.0T + 125T^{2} \) |
| 13 | \( 1 - 2.76T + 2.19e3T^{2} \) |
| 17 | \( 1 - 50.6T + 4.91e3T^{2} \) |
| 19 | \( 1 - 56.5T + 6.85e3T^{2} \) |
| 23 | \( 1 - 172.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 282.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 237.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 207.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 386.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 18.3T + 7.95e4T^{2} \) |
| 47 | \( 1 - 309.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 480.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 114.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 109.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 567.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 780.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 605.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 686.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 316.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.61e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.23e3T + 9.12e5T^{2} \) |
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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
−9.850077558308333407642651123101, −9.425518644258138302195217914962, −8.624046254762289435025889291261, −7.13329046136621270068806198138, −6.12572047941256166886702454952, −5.50980137732487071677128003588, −4.87260921949520813706180408123, −3.37103794082095769803280396569, −2.49900577162958189467731688533, −1.04405278275181453343337043252,
1.04405278275181453343337043252, 2.49900577162958189467731688533, 3.37103794082095769803280396569, 4.87260921949520813706180408123, 5.50980137732487071677128003588, 6.12572047941256166886702454952, 7.13329046136621270068806198138, 8.624046254762289435025889291261, 9.425518644258138302195217914962, 9.850077558308333407642651123101
