| L(s) = 1 | + 2-s + 2.90·3-s + 4-s + 5-s + 2.90·6-s − 4.42·7-s + 8-s + 5.42·9-s + 10-s − 2.28·11-s + 2.90·12-s − 4.90·13-s − 4.42·14-s + 2.90·15-s + 16-s − 4.42·17-s + 5.42·18-s + 7.05·19-s + 20-s − 12.8·21-s − 2.28·22-s − 0.622·23-s + 2.90·24-s + 25-s − 4.90·26-s + 7.05·27-s − 4.42·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.67·3-s + 0.5·4-s + 0.447·5-s + 1.18·6-s − 1.67·7-s + 0.353·8-s + 1.80·9-s + 0.316·10-s − 0.687·11-s + 0.838·12-s − 1.35·13-s − 1.18·14-s + 0.749·15-s + 0.250·16-s − 1.07·17-s + 1.27·18-s + 1.61·19-s + 0.223·20-s − 2.80·21-s − 0.486·22-s − 0.129·23-s + 0.592·24-s + 0.200·25-s − 0.961·26-s + 1.35·27-s − 0.836·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 310 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 310 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.807595300\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.807595300\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 31 | \( 1 - T \) |
| good | 3 | \( 1 - 2.90T + 3T^{2} \) |
| 7 | \( 1 + 4.42T + 7T^{2} \) |
| 11 | \( 1 + 2.28T + 11T^{2} \) |
| 13 | \( 1 + 4.90T + 13T^{2} \) |
| 17 | \( 1 + 4.42T + 17T^{2} \) |
| 19 | \( 1 - 7.05T + 19T^{2} \) |
| 23 | \( 1 + 0.622T + 23T^{2} \) |
| 29 | \( 1 + 2.76T + 29T^{2} \) |
| 37 | \( 1 - 3.95T + 37T^{2} \) |
| 41 | \( 1 - 3.67T + 41T^{2} \) |
| 43 | \( 1 - 7.76T + 43T^{2} \) |
| 47 | \( 1 - 11.1T + 47T^{2} \) |
| 53 | \( 1 + 0.0459T + 53T^{2} \) |
| 59 | \( 1 - 2.19T + 59T^{2} \) |
| 61 | \( 1 + 3.71T + 61T^{2} \) |
| 67 | \( 1 + 8.29T + 67T^{2} \) |
| 71 | \( 1 - 2.75T + 71T^{2} \) |
| 73 | \( 1 + 15.4T + 73T^{2} \) |
| 79 | \( 1 + 13.4T + 79T^{2} \) |
| 83 | \( 1 - 9.95T + 83T^{2} \) |
| 89 | \( 1 - 14.8T + 89T^{2} \) |
| 97 | \( 1 + 12.2T + 97T^{2} \) |
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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
−12.13193901211563313437503730072, −10.46316853637661289851115558783, −9.579275711568957330163033445572, −9.165139796077066041206006038709, −7.63943033515574615817185368192, −7.03305200213582884506676129561, −5.70567049517602572912149665902, −4.23551306753011003207867690219, −2.98188371330641182925130106096, −2.48838307944373786727591204572,
2.48838307944373786727591204572, 2.98188371330641182925130106096, 4.23551306753011003207867690219, 5.70567049517602572912149665902, 7.03305200213582884506676129561, 7.63943033515574615817185368192, 9.165139796077066041206006038709, 9.579275711568957330163033445572, 10.46316853637661289851115558783, 12.13193901211563313437503730072
