| L(s) = 1 | − 2.52·2-s + 3-s + 4.37·4-s − 2.37·5-s − 2.52·6-s − 0.792·7-s − 5.98·8-s + 9-s + 5.98·10-s + 4.37·12-s + 4.10·13-s + 2·14-s − 2.37·15-s + 6.37·16-s + 5.98·17-s − 2.52·18-s − 4.25·19-s − 10.3·20-s − 0.792·21-s + 2·23-s − 5.98·24-s + 0.627·25-s − 10.3·26-s + 27-s − 3.46·28-s + 2.52·29-s + 5.98·30-s + ⋯ |
| L(s) = 1 | − 1.78·2-s + 0.577·3-s + 2.18·4-s − 1.06·5-s − 1.03·6-s − 0.299·7-s − 2.11·8-s + 0.333·9-s + 1.89·10-s + 1.26·12-s + 1.13·13-s + 0.534·14-s − 0.612·15-s + 1.59·16-s + 1.45·17-s − 0.594·18-s − 0.976·19-s − 2.31·20-s − 0.172·21-s + 0.417·23-s − 1.22·24-s + 0.125·25-s − 2.03·26-s + 0.192·27-s − 0.654·28-s + 0.468·29-s + 1.09·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6192316075\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6192316075\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 2.52T + 2T^{2} \) |
| 5 | \( 1 + 2.37T + 5T^{2} \) |
| 7 | \( 1 + 0.792T + 7T^{2} \) |
| 13 | \( 1 - 4.10T + 13T^{2} \) |
| 17 | \( 1 - 5.98T + 17T^{2} \) |
| 19 | \( 1 + 4.25T + 19T^{2} \) |
| 23 | \( 1 - 2T + 23T^{2} \) |
| 29 | \( 1 - 2.52T + 29T^{2} \) |
| 31 | \( 1 - 7.37T + 31T^{2} \) |
| 37 | \( 1 - 5T + 37T^{2} \) |
| 41 | \( 1 - 5.69T + 41T^{2} \) |
| 43 | \( 1 - 6.63T + 43T^{2} \) |
| 47 | \( 1 + 1.25T + 47T^{2} \) |
| 53 | \( 1 - 13.1T + 53T^{2} \) |
| 59 | \( 1 + 6T + 59T^{2} \) |
| 61 | \( 1 - 2.67T + 61T^{2} \) |
| 67 | \( 1 - 16.1T + 67T^{2} \) |
| 71 | \( 1 - 0.744T + 71T^{2} \) |
| 73 | \( 1 + 7.42T + 73T^{2} \) |
| 79 | \( 1 + 5.84T + 79T^{2} \) |
| 83 | \( 1 + 8.51T + 83T^{2} \) |
| 89 | \( 1 + 6.37T + 89T^{2} \) |
| 97 | \( 1 + 12.4T + 97T^{2} \) |
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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
−11.14482558751290845869326472592, −10.30861232147378039791530806874, −9.478668154943856534217489353053, −8.447189326746436467389406982449, −8.082886010611771925236573919365, −7.17610448933405091698685460594, −6.13183667544622170210800127864, −4.01586908165467332267389728373, −2.78563178180183564246620900160, −1.01647366498656575723404249336,
1.01647366498656575723404249336, 2.78563178180183564246620900160, 4.01586908165467332267389728373, 6.13183667544622170210800127864, 7.17610448933405091698685460594, 8.082886010611771925236573919365, 8.447189326746436467389406982449, 9.478668154943856534217489353053, 10.30861232147378039791530806874, 11.14482558751290845869326472592
