| L(s) = 1 | − 19.7·2-s + 99.5·3-s − 121.·4-s + 1.52e3·5-s − 1.96e3·6-s + 9.24e3·7-s + 1.25e4·8-s − 9.78e3·9-s − 3.00e4·10-s − 3.88e4·11-s − 1.20e4·12-s − 1.82e5·14-s + 1.51e5·15-s − 1.85e5·16-s + 4.53e5·17-s + 1.93e5·18-s + 9.65e5·19-s − 1.84e5·20-s + 9.19e5·21-s + 7.67e5·22-s + 5.43e5·23-s + 1.24e6·24-s + 3.61e5·25-s − 2.93e6·27-s − 1.11e6·28-s + 4.05e6·29-s − 2.99e6·30-s + ⋯ |
| L(s) = 1 | − 0.873·2-s + 0.709·3-s − 0.236·4-s + 1.08·5-s − 0.619·6-s + 1.45·7-s + 1.08·8-s − 0.496·9-s − 0.951·10-s − 0.799·11-s − 0.167·12-s − 1.27·14-s + 0.772·15-s − 0.707·16-s + 1.31·17-s + 0.434·18-s + 1.69·19-s − 0.257·20-s + 1.03·21-s + 0.698·22-s + 0.404·23-s + 0.766·24-s + 0.185·25-s − 1.06·27-s − 0.344·28-s + 1.06·29-s − 0.674·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(2.455117623\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.455117623\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 \) |
| good | 2 | \( 1 + 19.7T + 512T^{2} \) |
| 3 | \( 1 - 99.5T + 1.96e4T^{2} \) |
| 5 | \( 1 - 1.52e3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 9.24e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 3.88e4T + 2.35e9T^{2} \) |
| 17 | \( 1 - 4.53e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 9.65e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 5.43e5T + 1.80e12T^{2} \) |
| 29 | \( 1 - 4.05e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 2.44e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 7.60e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 4.47e6T + 3.27e14T^{2} \) |
| 43 | \( 1 - 1.62e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 1.05e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 4.76e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 1.11e8T + 8.66e15T^{2} \) |
| 61 | \( 1 - 6.91e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 2.81e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 3.46e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 1.76e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 2.36e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 2.90e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 4.52e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 1.48e9T + 7.60e17T^{2} \) |
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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
−10.72100082557929497735642683818, −9.868319304561476701059034264150, −9.071815476826559103709257990423, −8.079132999443033250568151273876, −7.61207562459972767120200676932, −5.60224822504485735866749862539, −4.87414513228519324989871515941, −3.04020930470510010913640903704, −1.81358915357501837722808332949, −0.932941644921500772518140071669,
0.932941644921500772518140071669, 1.81358915357501837722808332949, 3.04020930470510010913640903704, 4.87414513228519324989871515941, 5.60224822504485735866749862539, 7.61207562459972767120200676932, 8.079132999443033250568151273876, 9.071815476826559103709257990423, 9.868319304561476701059034264150, 10.72100082557929497735642683818
