| L(s) = 1 | − 2.32·2-s − 1.61·3-s + 3.38·4-s + 5-s + 3.73·6-s − 7-s − 3.22·8-s − 0.406·9-s − 2.32·10-s − 1.92·11-s − 5.45·12-s − 2.65·13-s + 2.32·14-s − 1.61·15-s + 0.711·16-s − 1.71·17-s + 0.943·18-s − 6.17·19-s + 3.38·20-s + 1.61·21-s + 4.46·22-s − 3.52·23-s + 5.19·24-s + 25-s + 6.16·26-s + 5.48·27-s − 3.38·28-s + ⋯ |
| L(s) = 1 | − 1.64·2-s − 0.929·3-s + 1.69·4-s + 0.447·5-s + 1.52·6-s − 0.377·7-s − 1.14·8-s − 0.135·9-s − 0.734·10-s − 0.579·11-s − 1.57·12-s − 0.736·13-s + 0.620·14-s − 0.415·15-s + 0.177·16-s − 0.415·17-s + 0.222·18-s − 1.41·19-s + 0.758·20-s + 0.351·21-s + 0.951·22-s − 0.735·23-s + 1.06·24-s + 0.200·25-s + 1.20·26-s + 1.05·27-s − 0.640·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 229 | \( 1 - T \) |
| good | 2 | \( 1 + 2.32T + 2T^{2} \) |
| 3 | \( 1 + 1.61T + 3T^{2} \) |
| 11 | \( 1 + 1.92T + 11T^{2} \) |
| 13 | \( 1 + 2.65T + 13T^{2} \) |
| 17 | \( 1 + 1.71T + 17T^{2} \) |
| 19 | \( 1 + 6.17T + 19T^{2} \) |
| 23 | \( 1 + 3.52T + 23T^{2} \) |
| 29 | \( 1 + 3.98T + 29T^{2} \) |
| 31 | \( 1 - 0.382T + 31T^{2} \) |
| 37 | \( 1 - 1.71T + 37T^{2} \) |
| 41 | \( 1 + 3.97T + 41T^{2} \) |
| 43 | \( 1 - 4.07T + 43T^{2} \) |
| 47 | \( 1 - 12.2T + 47T^{2} \) |
| 53 | \( 1 - 0.976T + 53T^{2} \) |
| 59 | \( 1 - 13.5T + 59T^{2} \) |
| 61 | \( 1 - 4.81T + 61T^{2} \) |
| 67 | \( 1 - 10.9T + 67T^{2} \) |
| 71 | \( 1 + 14.7T + 71T^{2} \) |
| 73 | \( 1 - 10.8T + 73T^{2} \) |
| 79 | \( 1 - 13.3T + 79T^{2} \) |
| 83 | \( 1 - 9.80T + 83T^{2} \) |
| 89 | \( 1 - 16.3T + 89T^{2} \) |
| 97 | \( 1 - 11.0T + 97T^{2} \) |
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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
−7.59993670511995274439842556648, −6.80365302900779384906915909951, −6.36138325722750247847699764277, −5.63846374396997533772901909933, −4.91077511620935993266169305805, −3.89780008011023967932326395427, −2.43640631622011942719337159520, −2.19029984322181263263652974860, −0.76995048775247212165781887101, 0,
0.76995048775247212165781887101, 2.19029984322181263263652974860, 2.43640631622011942719337159520, 3.89780008011023967932326395427, 4.91077511620935993266169305805, 5.63846374396997533772901909933, 6.36138325722750247847699764277, 6.80365302900779384906915909951, 7.59993670511995274439842556648
