| L(s) = 1 | + 1.57·2-s − 3-s + 0.489·4-s − 5-s − 1.57·6-s − 4.24·7-s − 2.38·8-s + 9-s − 1.57·10-s − 0.930·11-s − 0.489·12-s − 6.69·14-s + 15-s − 4.73·16-s + 6.69·17-s + 1.57·18-s − 4.08·19-s − 0.489·20-s + 4.24·21-s − 1.46·22-s + 4.83·23-s + 2.38·24-s + 25-s − 27-s − 2.07·28-s + 5.09·29-s + 1.57·30-s + ⋯ |
| L(s) = 1 | + 1.11·2-s − 0.577·3-s + 0.244·4-s − 0.447·5-s − 0.644·6-s − 1.60·7-s − 0.842·8-s + 0.333·9-s − 0.498·10-s − 0.280·11-s − 0.141·12-s − 1.78·14-s + 0.258·15-s − 1.18·16-s + 1.62·17-s + 0.371·18-s − 0.936·19-s − 0.109·20-s + 0.925·21-s − 0.313·22-s + 1.00·23-s + 0.486·24-s + 0.200·25-s − 0.192·27-s − 0.392·28-s + 0.946·29-s + 0.288·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2535 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2535 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.321752883\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.321752883\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 1.57T + 2T^{2} \) |
| 7 | \( 1 + 4.24T + 7T^{2} \) |
| 11 | \( 1 + 0.930T + 11T^{2} \) |
| 17 | \( 1 - 6.69T + 17T^{2} \) |
| 19 | \( 1 + 4.08T + 19T^{2} \) |
| 23 | \( 1 - 4.83T + 23T^{2} \) |
| 29 | \( 1 - 5.09T + 29T^{2} \) |
| 31 | \( 1 + 10.0T + 31T^{2} \) |
| 37 | \( 1 + 6.44T + 37T^{2} \) |
| 41 | \( 1 - 10.1T + 41T^{2} \) |
| 43 | \( 1 - 4.26T + 43T^{2} \) |
| 47 | \( 1 - 3.33T + 47T^{2} \) |
| 53 | \( 1 + 6.11T + 53T^{2} \) |
| 59 | \( 1 - 1.90T + 59T^{2} \) |
| 61 | \( 1 - 4.34T + 61T^{2} \) |
| 67 | \( 1 - 10.0T + 67T^{2} \) |
| 71 | \( 1 - 9.28T + 71T^{2} \) |
| 73 | \( 1 - 15.2T + 73T^{2} \) |
| 79 | \( 1 - 2.83T + 79T^{2} \) |
| 83 | \( 1 + 0.543T + 83T^{2} \) |
| 89 | \( 1 - 12.4T + 89T^{2} \) |
| 97 | \( 1 + 1.73T + 97T^{2} \) |
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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
−9.116020530607023321305646314513, −8.001424966854283770017462469799, −6.99457474999403705691606371725, −6.45573967346259479777657552321, −5.62687590182582593809075864867, −5.11090869867130750425821928408, −3.95882945245364818462423740510, −3.48808504461606079442391643595, −2.61155729877292314092038011275, −0.60887829060028628393734114025,
0.60887829060028628393734114025, 2.61155729877292314092038011275, 3.48808504461606079442391643595, 3.95882945245364818462423740510, 5.11090869867130750425821928408, 5.62687590182582593809075864867, 6.45573967346259479777657552321, 6.99457474999403705691606371725, 8.001424966854283770017462469799, 9.116020530607023321305646314513
