| L(s) = 1 | − 3.40·3-s − 5-s − 2.33·7-s + 8.60·9-s + 6.16·11-s + 3.40·15-s + 3.07·17-s + 0.0902·19-s + 7.96·21-s + 4.81·23-s + 25-s − 19.0·27-s + 5.63·29-s − 8.34·31-s − 21.0·33-s + 2.33·35-s − 0.794·37-s + 3.95·41-s − 8.75·43-s − 8.60·45-s + 3.67·47-s − 1.53·49-s − 10.4·51-s + 8.08·53-s − 6.16·55-s − 0.307·57-s − 0.379·59-s + ⋯ |
| L(s) = 1 | − 1.96·3-s − 0.447·5-s − 0.883·7-s + 2.86·9-s + 1.85·11-s + 0.879·15-s + 0.745·17-s + 0.0207·19-s + 1.73·21-s + 1.00·23-s + 0.200·25-s − 3.67·27-s + 1.04·29-s − 1.49·31-s − 3.65·33-s + 0.395·35-s − 0.130·37-s + 0.618·41-s − 1.33·43-s − 1.28·45-s + 0.536·47-s − 0.219·49-s − 1.46·51-s + 1.11·53-s − 0.831·55-s − 0.0407·57-s − 0.0494·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8196668144\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8196668144\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + 3.40T + 3T^{2} \) |
| 7 | \( 1 + 2.33T + 7T^{2} \) |
| 11 | \( 1 - 6.16T + 11T^{2} \) |
| 17 | \( 1 - 3.07T + 17T^{2} \) |
| 19 | \( 1 - 0.0902T + 19T^{2} \) |
| 23 | \( 1 - 4.81T + 23T^{2} \) |
| 29 | \( 1 - 5.63T + 29T^{2} \) |
| 31 | \( 1 + 8.34T + 31T^{2} \) |
| 37 | \( 1 + 0.794T + 37T^{2} \) |
| 41 | \( 1 - 3.95T + 41T^{2} \) |
| 43 | \( 1 + 8.75T + 43T^{2} \) |
| 47 | \( 1 - 3.67T + 47T^{2} \) |
| 53 | \( 1 - 8.08T + 53T^{2} \) |
| 59 | \( 1 + 0.379T + 59T^{2} \) |
| 61 | \( 1 + 4.96T + 61T^{2} \) |
| 67 | \( 1 + 0.376T + 67T^{2} \) |
| 71 | \( 1 - 9.04T + 71T^{2} \) |
| 73 | \( 1 + 5.48T + 73T^{2} \) |
| 79 | \( 1 + 4.79T + 79T^{2} \) |
| 83 | \( 1 + 11.6T + 83T^{2} \) |
| 89 | \( 1 - 11.7T + 89T^{2} \) |
| 97 | \( 1 + 2.43T + 97T^{2} \) |
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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
−8.797975555858564994521396781086, −7.42742584525047477961042060176, −6.89886167555757737212228960119, −6.37869982901242399345846756468, −5.71885275653110581456076763228, −4.88341278241790198948625739816, −4.06970216895660769464798529052, −3.38911233934669049877242905290, −1.49609864538553886379069701153, −0.65170403552692261229580577739,
0.65170403552692261229580577739, 1.49609864538553886379069701153, 3.38911233934669049877242905290, 4.06970216895660769464798529052, 4.88341278241790198948625739816, 5.71885275653110581456076763228, 6.37869982901242399345846756468, 6.89886167555757737212228960119, 7.42742584525047477961042060176, 8.797975555858564994521396781086
