| L(s) = 1 | − 2-s + 3-s + 4-s − 1.56·5-s − 6-s − 1.09·7-s − 8-s + 9-s + 1.56·10-s + 4.17·11-s + 12-s − 2.33·13-s + 1.09·14-s − 1.56·15-s + 16-s − 0.0496·17-s − 18-s + 2.48·19-s − 1.56·20-s − 1.09·21-s − 4.17·22-s + 23-s − 24-s − 2.56·25-s + 2.33·26-s + 27-s − 1.09·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.698·5-s − 0.408·6-s − 0.412·7-s − 0.353·8-s + 0.333·9-s + 0.493·10-s + 1.25·11-s + 0.288·12-s − 0.648·13-s + 0.291·14-s − 0.403·15-s + 0.250·16-s − 0.0120·17-s − 0.235·18-s + 0.569·19-s − 0.349·20-s − 0.238·21-s − 0.890·22-s + 0.208·23-s − 0.204·24-s − 0.512·25-s + 0.458·26-s + 0.192·27-s − 0.206·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.383376459\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.383376459\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 5 | \( 1 + 1.56T + 5T^{2} \) |
| 7 | \( 1 + 1.09T + 7T^{2} \) |
| 11 | \( 1 - 4.17T + 11T^{2} \) |
| 13 | \( 1 + 2.33T + 13T^{2} \) |
| 17 | \( 1 + 0.0496T + 17T^{2} \) |
| 19 | \( 1 - 2.48T + 19T^{2} \) |
| 31 | \( 1 + 0.863T + 31T^{2} \) |
| 37 | \( 1 - 9.50T + 37T^{2} \) |
| 41 | \( 1 + 10.0T + 41T^{2} \) |
| 43 | \( 1 - 1.25T + 43T^{2} \) |
| 47 | \( 1 - 0.476T + 47T^{2} \) |
| 53 | \( 1 - 12.0T + 53T^{2} \) |
| 59 | \( 1 - 1.02T + 59T^{2} \) |
| 61 | \( 1 - 12.5T + 61T^{2} \) |
| 67 | \( 1 + 5.94T + 67T^{2} \) |
| 71 | \( 1 - 4.49T + 71T^{2} \) |
| 73 | \( 1 + 0.0841T + 73T^{2} \) |
| 79 | \( 1 + 12.4T + 79T^{2} \) |
| 83 | \( 1 - 2.88T + 83T^{2} \) |
| 89 | \( 1 - 6.00T + 89T^{2} \) |
| 97 | \( 1 + 1.94T + 97T^{2} \) |
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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
−8.483012883296788311330797106196, −7.79465497337517114363767956676, −7.11342669809506716150550606157, −6.59064063845597889790148194142, −5.59573972784434273316741651444, −4.44355445867998118212552951991, −3.70981552321667188876355036032, −2.95001307629233913367159236299, −1.89331894051472742213117147580, −0.73263615379500269157525798365,
0.73263615379500269157525798365, 1.89331894051472742213117147580, 2.95001307629233913367159236299, 3.70981552321667188876355036032, 4.44355445867998118212552951991, 5.59573972784434273316741651444, 6.59064063845597889790148194142, 7.11342669809506716150550606157, 7.79465497337517114363767956676, 8.483012883296788311330797106196
