| L(s) = 1 | − 2.46·2-s − 3-s + 4.09·4-s + 5-s + 2.46·6-s + 2.31·7-s − 5.15·8-s + 9-s − 2.46·10-s + 3.15·11-s − 4.09·12-s + 1.91·13-s − 5.71·14-s − 15-s + 4.55·16-s − 4.69·17-s − 2.46·18-s + 4·19-s + 4.09·20-s − 2.31·21-s − 7.78·22-s − 6.42·23-s + 5.15·24-s + 25-s − 4.72·26-s − 27-s + 9.46·28-s + ⋯ |
| L(s) = 1 | − 1.74·2-s − 0.577·3-s + 2.04·4-s + 0.447·5-s + 1.00·6-s + 0.874·7-s − 1.82·8-s + 0.333·9-s − 0.780·10-s + 0.950·11-s − 1.18·12-s + 0.531·13-s − 1.52·14-s − 0.258·15-s + 1.13·16-s − 1.13·17-s − 0.581·18-s + 0.917·19-s + 0.914·20-s − 0.505·21-s − 1.65·22-s − 1.33·23-s + 1.05·24-s + 0.200·25-s − 0.927·26-s − 0.192·27-s + 1.78·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6571975749\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6571975749\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 31 | \( 1 + T \) |
| good | 2 | \( 1 + 2.46T + 2T^{2} \) |
| 7 | \( 1 - 2.31T + 7T^{2} \) |
| 11 | \( 1 - 3.15T + 11T^{2} \) |
| 13 | \( 1 - 1.91T + 13T^{2} \) |
| 17 | \( 1 + 4.69T + 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + 6.42T + 23T^{2} \) |
| 29 | \( 1 - 0.776T + 29T^{2} \) |
| 37 | \( 1 - 6.62T + 37T^{2} \) |
| 41 | \( 1 - 8.71T + 41T^{2} \) |
| 43 | \( 1 + 2.23T + 43T^{2} \) |
| 47 | \( 1 - 11.8T + 47T^{2} \) |
| 53 | \( 1 - 2.24T + 53T^{2} \) |
| 59 | \( 1 - 1.97T + 59T^{2} \) |
| 61 | \( 1 + 1.07T + 61T^{2} \) |
| 67 | \( 1 - 8.40T + 67T^{2} \) |
| 71 | \( 1 - 12.1T + 71T^{2} \) |
| 73 | \( 1 + 0.749T + 73T^{2} \) |
| 79 | \( 1 + 13.8T + 79T^{2} \) |
| 83 | \( 1 + 1.93T + 83T^{2} \) |
| 89 | \( 1 + 0.195T + 89T^{2} \) |
| 97 | \( 1 + 5.15T + 97T^{2} \) |
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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
−11.04916228123310899836671935043, −9.998110961325233670542025853315, −9.286919969012647963335824469182, −8.481291345098533876852829991087, −7.57659076213834897420278428115, −6.62867761598136711058410123255, −5.78269881570778371135926794523, −4.25471806108173832146126025913, −2.17887318093600887399742421013, −1.07216669848970112173245795198,
1.07216669848970112173245795198, 2.17887318093600887399742421013, 4.25471806108173832146126025913, 5.78269881570778371135926794523, 6.62867761598136711058410123255, 7.57659076213834897420278428115, 8.481291345098533876852829991087, 9.286919969012647963335824469182, 9.998110961325233670542025853315, 11.04916228123310899836671935043
