| L(s) = 1 | − 0.876·2-s + 0.718·3-s − 1.23·4-s + 5-s − 0.629·6-s − 7-s + 2.83·8-s − 2.48·9-s − 0.876·10-s + 0.114·11-s − 0.886·12-s − 5.71·13-s + 0.876·14-s + 0.718·15-s − 0.0164·16-s + 3.29·17-s + 2.17·18-s − 0.312·19-s − 1.23·20-s − 0.718·21-s − 0.100·22-s + 9.00·23-s + 2.03·24-s + 25-s + 5.00·26-s − 3.94·27-s + 1.23·28-s + ⋯ |
| L(s) = 1 | − 0.619·2-s + 0.415·3-s − 0.616·4-s + 0.447·5-s − 0.257·6-s − 0.377·7-s + 1.00·8-s − 0.827·9-s − 0.277·10-s + 0.0345·11-s − 0.255·12-s − 1.58·13-s + 0.234·14-s + 0.185·15-s − 0.00412·16-s + 0.798·17-s + 0.512·18-s − 0.0717·19-s − 0.275·20-s − 0.156·21-s − 0.0213·22-s + 1.87·23-s + 0.415·24-s + 0.200·25-s + 0.981·26-s − 0.758·27-s + 0.232·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1295 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1295 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.036165910\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.036165910\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 37 | \( 1 + T \) |
| good | 2 | \( 1 + 0.876T + 2T^{2} \) |
| 3 | \( 1 - 0.718T + 3T^{2} \) |
| 11 | \( 1 - 0.114T + 11T^{2} \) |
| 13 | \( 1 + 5.71T + 13T^{2} \) |
| 17 | \( 1 - 3.29T + 17T^{2} \) |
| 19 | \( 1 + 0.312T + 19T^{2} \) |
| 23 | \( 1 - 9.00T + 23T^{2} \) |
| 29 | \( 1 - 9.18T + 29T^{2} \) |
| 31 | \( 1 - 2.16T + 31T^{2} \) |
| 41 | \( 1 - 6.64T + 41T^{2} \) |
| 43 | \( 1 + 0.971T + 43T^{2} \) |
| 47 | \( 1 + 0.611T + 47T^{2} \) |
| 53 | \( 1 - 6.82T + 53T^{2} \) |
| 59 | \( 1 + 7.75T + 59T^{2} \) |
| 61 | \( 1 + 12.7T + 61T^{2} \) |
| 67 | \( 1 - 0.659T + 67T^{2} \) |
| 71 | \( 1 - 15.1T + 71T^{2} \) |
| 73 | \( 1 - 10.1T + 73T^{2} \) |
| 79 | \( 1 - 11.3T + 79T^{2} \) |
| 83 | \( 1 - 13.1T + 83T^{2} \) |
| 89 | \( 1 + 9.92T + 89T^{2} \) |
| 97 | \( 1 - 5.10T + 97T^{2} \) |
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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
−9.436063934451231523245552114215, −9.067318014673511556938575878582, −8.144754952092126684817519319749, −7.46764113605273393920906134357, −6.48293448377171547420755557807, −5.28527279725748876567349042035, −4.71629270965591590662910469732, −3.28891754002365425844631562806, −2.44091000921260254928206960440, −0.806499269676483449242353947124,
0.806499269676483449242353947124, 2.44091000921260254928206960440, 3.28891754002365425844631562806, 4.71629270965591590662910469732, 5.28527279725748876567349042035, 6.48293448377171547420755557807, 7.46764113605273393920906134357, 8.144754952092126684817519319749, 9.067318014673511556938575878582, 9.436063934451231523245552114215
