| L(s) = 1 | − 4·2-s + 2·3-s + 16·4-s + 25·5-s − 8·6-s + 82·7-s − 64·8-s − 77·9-s − 100·10-s + 32·12-s − 328·14-s + 50·15-s + 256·16-s + 308·18-s + 400·20-s + 164·21-s − 878·23-s − 128·24-s + 625·25-s − 316·27-s + 1.31e3·28-s − 1.19e3·29-s − 200·30-s − 1.02e3·32-s + 2.05e3·35-s − 1.23e3·36-s − 1.60e3·40-s + ⋯ |
| L(s) = 1 | − 2-s + 2/9·3-s + 4-s + 5-s − 2/9·6-s + 1.67·7-s − 8-s − 0.950·9-s − 10-s + 2/9·12-s − 1.67·14-s + 2/9·15-s + 16-s + 0.950·18-s + 20-s + 0.371·21-s − 1.65·23-s − 2/9·24-s + 25-s − 0.433·27-s + 1.67·28-s − 1.42·29-s − 2/9·30-s − 32-s + 1.67·35-s − 0.950·36-s − 40-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.066118856\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.066118856\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{2} T \) |
| 5 | \( 1 - p^{2} T \) |
| good | 3 | \( 1 - 2 T + p^{4} T^{2} \) |
| 7 | \( 1 - 82 T + p^{4} T^{2} \) |
| 11 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 13 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 17 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 19 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 23 | \( 1 + 878 T + p^{4} T^{2} \) |
| 29 | \( 1 + 1198 T + p^{4} T^{2} \) |
| 31 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 37 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 41 | \( 1 - 482 T + p^{4} T^{2} \) |
| 43 | \( 1 + 2078 T + p^{4} T^{2} \) |
| 47 | \( 1 - 4402 T + p^{4} T^{2} \) |
| 53 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 59 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 61 | \( 1 + 4078 T + p^{4} T^{2} \) |
| 67 | \( 1 + 4478 T + p^{4} T^{2} \) |
| 71 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 73 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 79 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 83 | \( 1 - 8002 T + p^{4} T^{2} \) |
| 89 | \( 1 - 4322 T + p^{4} T^{2} \) |
| 97 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
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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
−17.64073418630670792245184839533, −16.80069546639948544895740131359, −14.94468297797950912592125522328, −13.98213889941486562114636196727, −11.78912304377010866929805253901, −10.61754994153513240072943887036, −9.046844846186826931840976115931, −7.86426940868673318362920130228, −5.73138720134652889788782692920, −1.98716358314195289745332918055,
1.98716358314195289745332918055, 5.73138720134652889788782692920, 7.86426940868673318362920130228, 9.046844846186826931840976115931, 10.61754994153513240072943887036, 11.78912304377010866929805253901, 13.98213889941486562114636196727, 14.94468297797950912592125522328, 16.80069546639948544895740131359, 17.64073418630670792245184839533
