| L(s) = 1 | − 2-s − 0.302·3-s + 4-s + 2.67·5-s + 0.302·6-s − 8-s − 2.90·9-s − 2.67·10-s − 4.53·11-s − 0.302·12-s + 5.61·13-s − 0.809·15-s + 16-s + 1.23·17-s + 2.90·18-s + 5.44·19-s + 2.67·20-s + 4.53·22-s + 4.32·23-s + 0.302·24-s + 2.15·25-s − 5.61·26-s + 1.78·27-s − 2.63·29-s + 0.809·30-s + 0.837·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.174·3-s + 0.5·4-s + 1.19·5-s + 0.123·6-s − 0.353·8-s − 0.969·9-s − 0.846·10-s − 1.36·11-s − 0.0873·12-s + 1.55·13-s − 0.208·15-s + 0.250·16-s + 0.299·17-s + 0.685·18-s + 1.24·19-s + 0.598·20-s + 0.966·22-s + 0.900·23-s + 0.0617·24-s + 0.431·25-s − 1.10·26-s + 0.343·27-s − 0.488·29-s + 0.147·30-s + 0.150·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.496848847\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.496848847\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 3 | \( 1 + 0.302T + 3T^{2} \) |
| 5 | \( 1 - 2.67T + 5T^{2} \) |
| 11 | \( 1 + 4.53T + 11T^{2} \) |
| 13 | \( 1 - 5.61T + 13T^{2} \) |
| 17 | \( 1 - 1.23T + 17T^{2} \) |
| 19 | \( 1 - 5.44T + 19T^{2} \) |
| 23 | \( 1 - 4.32T + 23T^{2} \) |
| 29 | \( 1 + 2.63T + 29T^{2} \) |
| 31 | \( 1 - 0.837T + 31T^{2} \) |
| 37 | \( 1 + 10.6T + 37T^{2} \) |
| 43 | \( 1 + 0.192T + 43T^{2} \) |
| 47 | \( 1 - 9.24T + 47T^{2} \) |
| 53 | \( 1 - 7.10T + 53T^{2} \) |
| 59 | \( 1 - 5.34T + 59T^{2} \) |
| 61 | \( 1 - 1.85T + 61T^{2} \) |
| 67 | \( 1 + 15.8T + 67T^{2} \) |
| 71 | \( 1 - 11.9T + 71T^{2} \) |
| 73 | \( 1 + 8.70T + 73T^{2} \) |
| 79 | \( 1 - 10.7T + 79T^{2} \) |
| 83 | \( 1 + 8.34T + 83T^{2} \) |
| 89 | \( 1 + 2.04T + 89T^{2} \) |
| 97 | \( 1 - 2.40T + 97T^{2} \) |
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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
−8.753298513419200206822986736102, −7.78306532727044050861006221863, −7.06477445753757918107979231702, −6.08189376341844310843262569160, −5.60154649202820523686107656026, −5.13466400802671422085680640158, −3.49685184631228161865213151070, −2.80920937528952476318077630752, −1.86432434692561275170696201252, −0.793638779721809011715806770661,
0.793638779721809011715806770661, 1.86432434692561275170696201252, 2.80920937528952476318077630752, 3.49685184631228161865213151070, 5.13466400802671422085680640158, 5.60154649202820523686107656026, 6.08189376341844310843262569160, 7.06477445753757918107979231702, 7.78306532727044050861006221863, 8.753298513419200206822986736102
