| L(s) = 1 | − 4·2-s − 1.16·3-s + 16·4-s − 65.1·5-s + 4.66·6-s + 94.5·7-s − 64·8-s − 241.·9-s + 260.·10-s − 458.·11-s − 18.6·12-s + 984.·13-s − 378.·14-s + 75.9·15-s + 256·16-s + 2.12e3·17-s + 966.·18-s − 1.04e3·20-s − 110.·21-s + 1.83e3·22-s − 3.84e3·23-s + 74.6·24-s + 1.11e3·25-s − 3.93e3·26-s + 565.·27-s + 1.51e3·28-s − 7.33e3·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.0747·3-s + 0.5·4-s − 1.16·5-s + 0.0528·6-s + 0.729·7-s − 0.353·8-s − 0.994·9-s + 0.823·10-s − 1.14·11-s − 0.0373·12-s + 1.61·13-s − 0.515·14-s + 0.0871·15-s + 0.250·16-s + 1.78·17-s + 0.703·18-s − 0.582·20-s − 0.0545·21-s + 0.807·22-s − 1.51·23-s + 0.0264·24-s + 0.357·25-s − 1.14·26-s + 0.149·27-s + 0.364·28-s − 1.61·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.7502629058\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7502629058\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 4T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + 1.16T + 243T^{2} \) |
| 5 | \( 1 + 65.1T + 3.12e3T^{2} \) |
| 7 | \( 1 - 94.5T + 1.68e4T^{2} \) |
| 11 | \( 1 + 458.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 984.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 2.12e3T + 1.41e6T^{2} \) |
| 23 | \( 1 + 3.84e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 7.33e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 1.17e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.76e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 2.45e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 2.12e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.27e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.31e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 9.23e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.02e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.86e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 6.87e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.62e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 2.19e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.02e5T + 3.93e9T^{2} \) |
| 89 | \( 1 + 2.82e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 8.07e4T + 8.58e9T^{2} \) |
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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
−9.620216569435727317291138914446, −8.366884036593754829174673214981, −8.094561024360658678091631546366, −7.51992795049438677900059115006, −6.02068873452004681903142128084, −5.38046829694585352149734998490, −3.90940379311908864825349177029, −3.11585818030737825126762399128, −1.68411876397693270862213419986, −0.45309394340730011117603970042,
0.45309394340730011117603970042, 1.68411876397693270862213419986, 3.11585818030737825126762399128, 3.90940379311908864825349177029, 5.38046829694585352149734998490, 6.02068873452004681903142128084, 7.51992795049438677900059115006, 8.094561024360658678091631546366, 8.366884036593754829174673214981, 9.620216569435727317291138914446
