| L(s) = 1 | + 2-s + 4-s − 3.23·5-s + 1.23·7-s + 8-s − 3.23·10-s − 11-s + 1.23·14-s + 16-s − 17-s − 3.23·20-s − 22-s − 2.76·23-s + 5.47·25-s + 1.23·28-s + 3.70·29-s + 3.23·31-s + 32-s − 34-s − 4.00·35-s − 3.70·37-s − 3.23·40-s − 4.47·41-s − 6.47·43-s − 44-s − 2.76·46-s − 3.52·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 1.44·5-s + 0.467·7-s + 0.353·8-s − 1.02·10-s − 0.301·11-s + 0.330·14-s + 0.250·16-s − 0.242·17-s − 0.723·20-s − 0.213·22-s − 0.576·23-s + 1.09·25-s + 0.233·28-s + 0.688·29-s + 0.581·31-s + 0.176·32-s − 0.171·34-s − 0.676·35-s − 0.609·37-s − 0.511·40-s − 0.698·41-s − 0.986·43-s − 0.150·44-s − 0.407·46-s − 0.514·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3366 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3366 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 + 3.23T + 5T^{2} \) |
| 7 | \( 1 - 1.23T + 7T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 2.76T + 23T^{2} \) |
| 29 | \( 1 - 3.70T + 29T^{2} \) |
| 31 | \( 1 - 3.23T + 31T^{2} \) |
| 37 | \( 1 + 3.70T + 37T^{2} \) |
| 41 | \( 1 + 4.47T + 41T^{2} \) |
| 43 | \( 1 + 6.47T + 43T^{2} \) |
| 47 | \( 1 + 3.52T + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 - 4.94T + 59T^{2} \) |
| 61 | \( 1 + 13.7T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 14.1T + 71T^{2} \) |
| 73 | \( 1 + 2T + 73T^{2} \) |
| 79 | \( 1 + 9.23T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 10.9T + 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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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
−8.206962382915080037381451157193, −7.47153912547534741162821590102, −6.82272794929600776232132882101, −5.93530118956427790222195884705, −4.88613777143795009663246934683, −4.45568541578556100794268692333, −3.58514714791821257757980544405, −2.85004833929362814309647818634, −1.56715672458721747146488272283, 0,
1.56715672458721747146488272283, 2.85004833929362814309647818634, 3.58514714791821257757980544405, 4.45568541578556100794268692333, 4.88613777143795009663246934683, 5.93530118956427790222195884705, 6.82272794929600776232132882101, 7.47153912547534741162821590102, 8.206962382915080037381451157193
