| L(s) = 1 | − 1.82·2-s − 3-s + 1.33·4-s + 3.12·5-s + 1.82·6-s − 7-s + 1.20·8-s + 9-s − 5.71·10-s + 5.32·11-s − 1.33·12-s − 6.23·13-s + 1.82·14-s − 3.12·15-s − 4.88·16-s + 3.42·17-s − 1.82·18-s + 3.00·19-s + 4.19·20-s + 21-s − 9.72·22-s − 3.54·23-s − 1.20·24-s + 4.78·25-s + 11.3·26-s − 27-s − 1.33·28-s + ⋯ |
| L(s) = 1 | − 1.29·2-s − 0.577·3-s + 0.669·4-s + 1.39·5-s + 0.746·6-s − 0.377·7-s + 0.426·8-s + 0.333·9-s − 1.80·10-s + 1.60·11-s − 0.386·12-s − 1.72·13-s + 0.488·14-s − 0.807·15-s − 1.22·16-s + 0.831·17-s − 0.430·18-s + 0.689·19-s + 0.937·20-s + 0.218·21-s − 2.07·22-s − 0.738·23-s − 0.246·24-s + 0.957·25-s + 2.23·26-s − 0.192·27-s − 0.253·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.003358285\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.003358285\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 191 | \( 1 - T \) |
| good | 2 | \( 1 + 1.82T + 2T^{2} \) |
| 5 | \( 1 - 3.12T + 5T^{2} \) |
| 11 | \( 1 - 5.32T + 11T^{2} \) |
| 13 | \( 1 + 6.23T + 13T^{2} \) |
| 17 | \( 1 - 3.42T + 17T^{2} \) |
| 19 | \( 1 - 3.00T + 19T^{2} \) |
| 23 | \( 1 + 3.54T + 23T^{2} \) |
| 29 | \( 1 - 9.68T + 29T^{2} \) |
| 31 | \( 1 + 7.96T + 31T^{2} \) |
| 37 | \( 1 - 3.64T + 37T^{2} \) |
| 41 | \( 1 - 0.294T + 41T^{2} \) |
| 43 | \( 1 + 6.81T + 43T^{2} \) |
| 47 | \( 1 + 6.75T + 47T^{2} \) |
| 53 | \( 1 - 13.0T + 53T^{2} \) |
| 59 | \( 1 - 7.44T + 59T^{2} \) |
| 61 | \( 1 - 5.24T + 61T^{2} \) |
| 67 | \( 1 + 1.88T + 67T^{2} \) |
| 71 | \( 1 - 12.6T + 71T^{2} \) |
| 73 | \( 1 - 4.58T + 73T^{2} \) |
| 79 | \( 1 - 8.40T + 79T^{2} \) |
| 83 | \( 1 - 11.1T + 83T^{2} \) |
| 89 | \( 1 + 3.49T + 89T^{2} \) |
| 97 | \( 1 + 12.4T + 97T^{2} \) |
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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
−8.646917016459003463095030108983, −7.75953797824081865863493847456, −6.85612592950266091299583837017, −6.58778939230092527331706292332, −5.54721920626884420887424756190, −4.94735034454370790090619703147, −3.84478348670103282122395084470, −2.48102166966482769344208567619, −1.64570447935457669238282761582, −0.75950770803331695400841198091,
0.75950770803331695400841198091, 1.64570447935457669238282761582, 2.48102166966482769344208567619, 3.84478348670103282122395084470, 4.94735034454370790090619703147, 5.54721920626884420887424756190, 6.58778939230092527331706292332, 6.85612592950266091299583837017, 7.75953797824081865863493847456, 8.646917016459003463095030108983
