| L(s) = 1 | + 45.5·2-s + 469.·3-s + 23.0·4-s + 4.40e3·5-s + 2.13e4·6-s + 6.30e4·7-s − 9.21e4·8-s + 4.30e4·9-s + 2.00e5·10-s + 3.45e5·11-s + 1.08e4·12-s − 3.71e5·13-s + 2.86e6·14-s + 2.06e6·15-s − 4.24e6·16-s + 6.22e6·17-s + 1.96e6·18-s − 4.76e6·19-s + 1.01e5·20-s + 2.95e7·21-s + 1.57e7·22-s − 3.03e7·23-s − 4.32e7·24-s − 2.94e7·25-s − 1.68e7·26-s − 6.29e7·27-s + 1.45e6·28-s + ⋯ |
| L(s) = 1 | + 1.00·2-s + 1.11·3-s + 0.0112·4-s + 0.630·5-s + 1.12·6-s + 1.41·7-s − 0.994·8-s + 0.243·9-s + 0.634·10-s + 0.645·11-s + 0.0125·12-s − 0.277·13-s + 1.42·14-s + 0.703·15-s − 1.01·16-s + 1.06·17-s + 0.244·18-s − 0.441·19-s + 0.00711·20-s + 1.57·21-s + 0.649·22-s − 0.981·23-s − 1.10·24-s − 0.602·25-s − 0.278·26-s − 0.843·27-s + 0.0159·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(4.088313332\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.088313332\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + 3.71e5T \) |
| good | 2 | \( 1 - 45.5T + 2.04e3T^{2} \) |
| 3 | \( 1 - 469.T + 1.77e5T^{2} \) |
| 5 | \( 1 - 4.40e3T + 4.88e7T^{2} \) |
| 7 | \( 1 - 6.30e4T + 1.97e9T^{2} \) |
| 11 | \( 1 - 3.45e5T + 2.85e11T^{2} \) |
| 17 | \( 1 - 6.22e6T + 3.42e13T^{2} \) |
| 19 | \( 1 + 4.76e6T + 1.16e14T^{2} \) |
| 23 | \( 1 + 3.03e7T + 9.52e14T^{2} \) |
| 29 | \( 1 + 2.00e8T + 1.22e16T^{2} \) |
| 31 | \( 1 + 3.08e7T + 2.54e16T^{2} \) |
| 37 | \( 1 - 5.78e8T + 1.77e17T^{2} \) |
| 41 | \( 1 + 3.26e8T + 5.50e17T^{2} \) |
| 43 | \( 1 + 9.40e8T + 9.29e17T^{2} \) |
| 47 | \( 1 - 1.21e9T + 2.47e18T^{2} \) |
| 53 | \( 1 - 1.04e9T + 9.26e18T^{2} \) |
| 59 | \( 1 - 7.61e6T + 3.01e19T^{2} \) |
| 61 | \( 1 - 9.96e9T + 4.35e19T^{2} \) |
| 67 | \( 1 - 5.96e9T + 1.22e20T^{2} \) |
| 71 | \( 1 + 1.16e10T + 2.31e20T^{2} \) |
| 73 | \( 1 - 2.52e10T + 3.13e20T^{2} \) |
| 79 | \( 1 + 2.27e10T + 7.47e20T^{2} \) |
| 83 | \( 1 - 1.90e10T + 1.28e21T^{2} \) |
| 89 | \( 1 - 9.90e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + 1.32e11T + 7.15e21T^{2} \) |
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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
−17.21920223914873204837171281149, −14.80637184764786391032747248332, −14.43405451618067474507693202822, −13.38854557390356835159382873043, −11.73078782241786425820245861066, −9.456313303724879965783512352234, −8.062890754944181565973480113734, −5.56782114283243308764359427356, −3.88399110217807465317239042256, −2.04534072595035030750174110108,
2.04534072595035030750174110108, 3.88399110217807465317239042256, 5.56782114283243308764359427356, 8.062890754944181565973480113734, 9.456313303724879965783512352234, 11.73078782241786425820245861066, 13.38854557390356835159382873043, 14.43405451618067474507693202822, 14.80637184764786391032747248332, 17.21920223914873204837171281149
