| L(s) = 1 | + 29.8·2-s − 70.1·3-s + 381.·4-s + 1.11e3·5-s − 2.09e3·6-s + 2.40e3·7-s − 3.89e3·8-s − 1.47e4·9-s + 3.32e4·10-s − 3.34e4·11-s − 2.67e4·12-s + 2.85e4·13-s + 7.17e4·14-s − 7.81e4·15-s − 3.11e5·16-s − 1.57e5·17-s − 4.41e5·18-s + 3.09e5·19-s + 4.25e5·20-s − 1.68e5·21-s − 1.00e6·22-s − 7.96e5·23-s + 2.73e5·24-s − 7.12e5·25-s + 8.53e5·26-s + 2.41e6·27-s + 9.16e5·28-s + ⋯ |
| L(s) = 1 | + 1.32·2-s − 0.499·3-s + 0.745·4-s + 0.796·5-s − 0.660·6-s + 0.377·7-s − 0.336·8-s − 0.750·9-s + 1.05·10-s − 0.689·11-s − 0.372·12-s + 0.277·13-s + 0.499·14-s − 0.398·15-s − 1.18·16-s − 0.456·17-s − 0.990·18-s + 0.545·19-s + 0.594·20-s − 0.188·21-s − 0.911·22-s − 0.593·23-s + 0.168·24-s − 0.364·25-s + 0.366·26-s + 0.874·27-s + 0.281·28-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
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
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 | \( 1 - 29.8T + 512T^{2} \) |
| 3 | \( 1 + 70.1T + 1.96e4T^{2} \) |
| 5 | \( 1 - 1.11e3T + 1.95e6T^{2} \) |
| 11 | \( 1 + 3.34e4T + 2.35e9T^{2} \) |
| 17 | \( 1 + 1.57e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 3.09e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 7.96e5T + 1.80e12T^{2} \) |
| 29 | \( 1 + 1.10e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 2.97e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 4.80e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + 2.00e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 3.06e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 6.06e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 4.36e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 3.11e6T + 8.66e15T^{2} \) |
| 61 | \( 1 - 3.50e6T + 1.16e16T^{2} \) |
| 67 | \( 1 - 2.57e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 1.34e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 2.83e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 1.55e7T + 1.19e17T^{2} \) |
| 83 | \( 1 - 4.31e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 5.70e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 2.89e7T + 7.60e17T^{2} \) |
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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
−11.84218425476401719833006929868, −11.06345546831847084575418409227, −9.689616027394414620788402895983, −8.296275663040301795601268568107, −6.52949889327216136585002957629, −5.60427916233106354976299748579, −4.90335598732201873427974249121, −3.33734785931127690206794973386, −2.02451089522675186898418455731, 0,
2.02451089522675186898418455731, 3.33734785931127690206794973386, 4.90335598732201873427974249121, 5.60427916233106354976299748579, 6.52949889327216136585002957629, 8.296275663040301795601268568107, 9.689616027394414620788402895983, 11.06345546831847084575418409227, 11.84218425476401719833006929868
