| L(s) = 1 | − 2.78·2-s + 3-s + 5.75·4-s − 1.59·5-s − 2.78·6-s + 7-s − 10.4·8-s + 9-s + 4.43·10-s + 0.134·11-s + 5.75·12-s − 2.78·14-s − 1.59·15-s + 17.5·16-s − 7.07·17-s − 2.78·18-s − 5.94·19-s − 9.15·20-s + 21-s − 0.375·22-s − 7.87·23-s − 10.4·24-s − 2.46·25-s + 27-s + 5.75·28-s + 7.91·29-s + 4.43·30-s + ⋯ |
| L(s) = 1 | − 1.96·2-s + 0.577·3-s + 2.87·4-s − 0.711·5-s − 1.13·6-s + 0.377·7-s − 3.69·8-s + 0.333·9-s + 1.40·10-s + 0.0406·11-s + 1.65·12-s − 0.744·14-s − 0.410·15-s + 4.39·16-s − 1.71·17-s − 0.656·18-s − 1.36·19-s − 2.04·20-s + 0.218·21-s − 0.0800·22-s − 1.64·23-s − 2.13·24-s − 0.493·25-s + 0.192·27-s + 1.08·28-s + 1.46·29-s + 0.808·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5741261425\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5741261425\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 2.78T + 2T^{2} \) |
| 5 | \( 1 + 1.59T + 5T^{2} \) |
| 11 | \( 1 - 0.134T + 11T^{2} \) |
| 17 | \( 1 + 7.07T + 17T^{2} \) |
| 19 | \( 1 + 5.94T + 19T^{2} \) |
| 23 | \( 1 + 7.87T + 23T^{2} \) |
| 29 | \( 1 - 7.91T + 29T^{2} \) |
| 31 | \( 1 - 2.66T + 31T^{2} \) |
| 37 | \( 1 - 0.101T + 37T^{2} \) |
| 41 | \( 1 - 3.72T + 41T^{2} \) |
| 43 | \( 1 - 0.825T + 43T^{2} \) |
| 47 | \( 1 + 2.58T + 47T^{2} \) |
| 53 | \( 1 - 1.42T + 53T^{2} \) |
| 59 | \( 1 - 7.93T + 59T^{2} \) |
| 61 | \( 1 + 3.77T + 61T^{2} \) |
| 67 | \( 1 + 4.56T + 67T^{2} \) |
| 71 | \( 1 - 10.6T + 71T^{2} \) |
| 73 | \( 1 + 1.49T + 73T^{2} \) |
| 79 | \( 1 - 13.0T + 79T^{2} \) |
| 83 | \( 1 - 15.6T + 83T^{2} \) |
| 89 | \( 1 - 7.17T + 89T^{2} \) |
| 97 | \( 1 - 13.9T + 97T^{2} \) |
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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
−8.483210367863528357719655783016, −8.106373685958040624147336166923, −7.50034664971122422477609560267, −6.56366887866650208072737080663, −6.21124472716498344948138027527, −4.56487253546816110879124051426, −3.66624984086382002183307980014, −2.38545744024980788122989818168, −2.00078150130570216010067376083, −0.55335323917017137252666975393,
0.55335323917017137252666975393, 2.00078150130570216010067376083, 2.38545744024980788122989818168, 3.66624984086382002183307980014, 4.56487253546816110879124051426, 6.21124472716498344948138027527, 6.56366887866650208072737080663, 7.50034664971122422477609560267, 8.106373685958040624147336166923, 8.483210367863528357719655783016
