| L(s) = 1 | − 19.0·2-s − 267.·3-s − 150.·4-s + 1.10e3·5-s + 5.08e3·6-s + 2.40e3·7-s + 1.25e4·8-s + 5.18e4·9-s − 2.10e4·10-s + 4.21e4·11-s + 4.03e4·12-s − 2.85e4·13-s − 4.56e4·14-s − 2.96e5·15-s − 1.62e5·16-s − 5.71e4·17-s − 9.85e5·18-s + 9.01e5·19-s − 1.66e5·20-s − 6.42e5·21-s − 8.01e5·22-s + 1.99e5·23-s − 3.36e6·24-s − 7.27e5·25-s + 5.42e5·26-s − 8.60e6·27-s − 3.62e5·28-s + ⋯ |
| L(s) = 1 | − 0.839·2-s − 1.90·3-s − 0.294·4-s + 0.792·5-s + 1.60·6-s + 0.377·7-s + 1.08·8-s + 2.63·9-s − 0.665·10-s + 0.868·11-s + 0.561·12-s − 0.277·13-s − 0.317·14-s − 1.50·15-s − 0.618·16-s − 0.165·17-s − 2.21·18-s + 1.58·19-s − 0.233·20-s − 0.720·21-s − 0.729·22-s + 0.148·23-s − 2.07·24-s − 0.372·25-s + 0.232·26-s − 3.11·27-s − 0.111·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.7854553615\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7854553615\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - 2.40e3T \) |
| 13 | \( 1 + 2.85e4T \) |
| good | 2 | \( 1 + 19.0T + 512T^{2} \) |
| 3 | \( 1 + 267.T + 1.96e4T^{2} \) |
| 5 | \( 1 - 1.10e3T + 1.95e6T^{2} \) |
| 11 | \( 1 - 4.21e4T + 2.35e9T^{2} \) |
| 17 | \( 1 + 5.71e4T + 1.18e11T^{2} \) |
| 19 | \( 1 - 9.01e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 1.99e5T + 1.80e12T^{2} \) |
| 29 | \( 1 - 2.30e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 2.67e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.43e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 2.31e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 2.17e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 1.73e6T + 1.11e15T^{2} \) |
| 53 | \( 1 + 7.68e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 1.47e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.41e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 9.37e7T + 2.72e16T^{2} \) |
| 71 | \( 1 + 3.59e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 4.24e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 1.29e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 5.02e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 4.03e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 1.35e9T + 7.60e17T^{2} \) |
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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
−11.90985798216773774819481241957, −11.10938600246390115929978304987, −9.998764267520858055308569685619, −9.428143165675907335060286351207, −7.64426629301253489158368153707, −6.45658782626833424102363816897, −5.36674243660923661592041621240, −4.41377284913811034416800818770, −1.49651914945754100230405644580, −0.71565627793444312550521160865,
0.71565627793444312550521160865, 1.49651914945754100230405644580, 4.41377284913811034416800818770, 5.36674243660923661592041621240, 6.45658782626833424102363816897, 7.64426629301253489158368153707, 9.428143165675907335060286351207, 9.998764267520858055308569685619, 11.10938600246390115929978304987, 11.90985798216773774819481241957
