| L(s) = 1 | + 0.265·2-s + 3-s − 1.92·4-s + 3.84·5-s + 0.265·6-s − 2.81·7-s − 1.04·8-s + 9-s + 1.02·10-s − 1.92·12-s − 13-s − 0.746·14-s + 3.84·15-s + 3.58·16-s + 3.48·17-s + 0.265·18-s − 4.69·19-s − 7.41·20-s − 2.81·21-s − 8.27·23-s − 1.04·24-s + 9.76·25-s − 0.265·26-s + 27-s + 5.42·28-s + 2.53·29-s + 1.02·30-s + ⋯ |
| L(s) = 1 | + 0.187·2-s + 0.577·3-s − 0.964·4-s + 1.71·5-s + 0.108·6-s − 1.06·7-s − 0.368·8-s + 0.333·9-s + 0.322·10-s − 0.556·12-s − 0.277·13-s − 0.199·14-s + 0.992·15-s + 0.895·16-s + 0.845·17-s + 0.0626·18-s − 1.07·19-s − 1.65·20-s − 0.613·21-s − 1.72·23-s − 0.213·24-s + 1.95·25-s − 0.0520·26-s + 0.192·27-s + 1.02·28-s + 0.471·29-s + 0.186·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4719 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4719 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.494975819\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.494975819\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 - 0.265T + 2T^{2} \) |
| 5 | \( 1 - 3.84T + 5T^{2} \) |
| 7 | \( 1 + 2.81T + 7T^{2} \) |
| 17 | \( 1 - 3.48T + 17T^{2} \) |
| 19 | \( 1 + 4.69T + 19T^{2} \) |
| 23 | \( 1 + 8.27T + 23T^{2} \) |
| 29 | \( 1 - 2.53T + 29T^{2} \) |
| 31 | \( 1 - 4.77T + 31T^{2} \) |
| 37 | \( 1 - 9.69T + 37T^{2} \) |
| 41 | \( 1 - 10.5T + 41T^{2} \) |
| 43 | \( 1 + 7.76T + 43T^{2} \) |
| 47 | \( 1 - 5.60T + 47T^{2} \) |
| 53 | \( 1 - 10.1T + 53T^{2} \) |
| 59 | \( 1 - 1.94T + 59T^{2} \) |
| 61 | \( 1 - 0.538T + 61T^{2} \) |
| 67 | \( 1 - 0.204T + 67T^{2} \) |
| 71 | \( 1 - 9.07T + 71T^{2} \) |
| 73 | \( 1 + 13.7T + 73T^{2} \) |
| 79 | \( 1 - 12.6T + 79T^{2} \) |
| 83 | \( 1 - 17.7T + 83T^{2} \) |
| 89 | \( 1 - 11.4T + 89T^{2} \) |
| 97 | \( 1 + 14.0T + 97T^{2} \) |
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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
−8.415443448259382099636470951541, −7.75409425055259260170719908416, −6.51161362118131227833988622420, −6.11975305543854193934322697705, −5.48919351355052975056053864594, −4.52345049813859412054072952642, −3.77703730977245528933692643127, −2.78765348176840787766292555082, −2.15035986689555763870957625676, −0.832286023540569262270989696263,
0.832286023540569262270989696263, 2.15035986689555763870957625676, 2.78765348176840787766292555082, 3.77703730977245528933692643127, 4.52345049813859412054072952642, 5.48919351355052975056053864594, 6.11975305543854193934322697705, 6.51161362118131227833988622420, 7.75409425055259260170719908416, 8.415443448259382099636470951541
