| L(s) = 1 | − 2-s + 2.66·3-s + 4-s − 4.12·5-s − 2.66·6-s − 4.21·7-s − 8-s + 4.12·9-s + 4.12·10-s − 11-s + 2.66·12-s − 2.21·13-s + 4.21·14-s − 11.0·15-s + 16-s − 3.45·17-s − 4.12·18-s − 19-s − 4.12·20-s − 11.2·21-s + 22-s − 5.45·23-s − 2.66·24-s + 12.0·25-s + 2.21·26-s + 3.00·27-s − 4.21·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.54·3-s + 0.5·4-s − 1.84·5-s − 1.08·6-s − 1.59·7-s − 0.353·8-s + 1.37·9-s + 1.30·10-s − 0.301·11-s + 0.770·12-s − 0.614·13-s + 1.12·14-s − 2.84·15-s + 0.250·16-s − 0.837·17-s − 0.972·18-s − 0.229·19-s − 0.922·20-s − 2.45·21-s + 0.213·22-s − 1.13·23-s − 0.544·24-s + 2.40·25-s + 0.434·26-s + 0.577·27-s − 0.796·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 2.66T + 3T^{2} \) |
| 5 | \( 1 + 4.12T + 5T^{2} \) |
| 7 | \( 1 + 4.21T + 7T^{2} \) |
| 13 | \( 1 + 2.21T + 13T^{2} \) |
| 17 | \( 1 + 3.45T + 17T^{2} \) |
| 23 | \( 1 + 5.45T + 23T^{2} \) |
| 29 | \( 1 - 5.57T + 29T^{2} \) |
| 31 | \( 1 - 7.00T + 31T^{2} \) |
| 37 | \( 1 + 2.90T + 37T^{2} \) |
| 41 | \( 1 + 11.9T + 41T^{2} \) |
| 43 | \( 1 - 1.46T + 43T^{2} \) |
| 47 | \( 1 - 7.58T + 47T^{2} \) |
| 53 | \( 1 + 13.2T + 53T^{2} \) |
| 59 | \( 1 - 4.79T + 59T^{2} \) |
| 61 | \( 1 - 8.90T + 61T^{2} \) |
| 67 | \( 1 - 1.30T + 67T^{2} \) |
| 71 | \( 1 + 6.80T + 71T^{2} \) |
| 73 | \( 1 + 1.45T + 73T^{2} \) |
| 79 | \( 1 + 9.15T + 79T^{2} \) |
| 83 | \( 1 + 13.2T + 83T^{2} \) |
| 89 | \( 1 - 8.24T + 89T^{2} \) |
| 97 | \( 1 - 2.18T + 97T^{2} \) |
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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
−10.40084940776631647005763218701, −9.718630081459298121388729713651, −8.658101307100522332694649716855, −8.237344083068515574965918741797, −7.30710961509607260717210449096, −6.60715229393566145156603212877, −4.33624233421165427434713181343, −3.38906630830624826018373564927, −2.62880618758018956896955195939, 0,
2.62880618758018956896955195939, 3.38906630830624826018373564927, 4.33624233421165427434713181343, 6.60715229393566145156603212877, 7.30710961509607260717210449096, 8.237344083068515574965918741797, 8.658101307100522332694649716855, 9.718630081459298121388729713651, 10.40084940776631647005763218701
