| L(s) = 1 | − 9.19·2-s − 81·3-s − 427.·4-s − 1.96e3·5-s + 744.·6-s + 1.19e4·7-s + 8.63e3·8-s + 6.56e3·9-s + 1.80e4·10-s − 1.31e4·11-s + 3.46e4·12-s − 4.23e4·13-s − 1.09e5·14-s + 1.58e5·15-s + 1.39e5·16-s + 4.30e5·17-s − 6.03e4·18-s − 2.24e5·19-s + 8.38e5·20-s − 9.65e5·21-s + 1.20e5·22-s − 3.39e4·23-s − 6.99e5·24-s + 1.89e6·25-s + 3.89e5·26-s − 5.31e5·27-s − 5.09e6·28-s + ⋯ |
| L(s) = 1 | − 0.406·2-s − 0.577·3-s − 0.834·4-s − 1.40·5-s + 0.234·6-s + 1.87·7-s + 0.745·8-s + 0.333·9-s + 0.570·10-s − 0.270·11-s + 0.482·12-s − 0.411·13-s − 0.762·14-s + 0.810·15-s + 0.532·16-s + 1.25·17-s − 0.135·18-s − 0.395·19-s + 1.17·20-s − 1.08·21-s + 0.109·22-s − 0.0252·23-s − 0.430·24-s + 0.970·25-s + 0.167·26-s − 0.192·27-s − 1.56·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.7644679400\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7644679400\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 81T \) |
| 59 | \( 1 - 1.21e7T \) |
| good | 2 | \( 1 + 9.19T + 512T^{2} \) |
| 5 | \( 1 + 1.96e3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 1.19e4T + 4.03e7T^{2} \) |
| 11 | \( 1 + 1.31e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 4.23e4T + 1.06e10T^{2} \) |
| 17 | \( 1 - 4.30e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 2.24e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 3.39e4T + 1.80e12T^{2} \) |
| 29 | \( 1 + 3.34e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 4.19e5T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.22e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 2.21e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 2.53e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 2.03e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 6.12e6T + 3.29e15T^{2} \) |
| 61 | \( 1 + 1.65e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 2.84e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 6.82e7T + 4.58e16T^{2} \) |
| 73 | \( 1 - 2.97e7T + 5.88e16T^{2} \) |
| 79 | \( 1 + 1.80e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 4.94e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 9.96e7T + 3.50e17T^{2} \) |
| 97 | \( 1 - 1.25e9T + 7.60e17T^{2} \) |
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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
−11.07648110909979306106411188816, −10.18716313996501228834716811326, −8.729949638853475102841597775263, −7.903955783764356967452224505308, −7.44577618492266069780383739244, −5.35189486736830326783081653350, −4.68424024392056515503890429325, −3.73795954075610386113771257599, −1.59150000288343342815906184547, −0.50669541643851450821591996781,
0.50669541643851450821591996781, 1.59150000288343342815906184547, 3.73795954075610386113771257599, 4.68424024392056515503890429325, 5.35189486736830326783081653350, 7.44577618492266069780383739244, 7.903955783764356967452224505308, 8.729949638853475102841597775263, 10.18716313996501228834716811326, 11.07648110909979306106411188816
