| L(s) = 1 | − 2-s + 1.23·3-s + 4-s − 5-s − 1.23·6-s + 3.50·7-s − 8-s − 1.48·9-s + 10-s − 1.86·11-s + 1.23·12-s − 6.67·13-s − 3.50·14-s − 1.23·15-s + 16-s + 6.33·17-s + 1.48·18-s + 4.59·19-s − 20-s + 4.31·21-s + 1.86·22-s − 8.71·23-s − 1.23·24-s + 25-s + 6.67·26-s − 5.51·27-s + 3.50·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.710·3-s + 0.5·4-s − 0.447·5-s − 0.502·6-s + 1.32·7-s − 0.353·8-s − 0.495·9-s + 0.316·10-s − 0.562·11-s + 0.355·12-s − 1.85·13-s − 0.936·14-s − 0.317·15-s + 0.250·16-s + 1.53·17-s + 0.350·18-s + 1.05·19-s − 0.223·20-s + 0.941·21-s + 0.397·22-s − 1.81·23-s − 0.251·24-s + 0.200·25-s + 1.30·26-s − 1.06·27-s + 0.662·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 401 | \( 1 + T \) |
| good | 3 | \( 1 - 1.23T + 3T^{2} \) |
| 7 | \( 1 - 3.50T + 7T^{2} \) |
| 11 | \( 1 + 1.86T + 11T^{2} \) |
| 13 | \( 1 + 6.67T + 13T^{2} \) |
| 17 | \( 1 - 6.33T + 17T^{2} \) |
| 19 | \( 1 - 4.59T + 19T^{2} \) |
| 23 | \( 1 + 8.71T + 23T^{2} \) |
| 29 | \( 1 - 9.04T + 29T^{2} \) |
| 31 | \( 1 + 8.47T + 31T^{2} \) |
| 37 | \( 1 + 3.07T + 37T^{2} \) |
| 41 | \( 1 + 3.18T + 41T^{2} \) |
| 43 | \( 1 + 2.83T + 43T^{2} \) |
| 47 | \( 1 - 8.62T + 47T^{2} \) |
| 53 | \( 1 - 6.47T + 53T^{2} \) |
| 59 | \( 1 + 5.42T + 59T^{2} \) |
| 61 | \( 1 + 4.38T + 61T^{2} \) |
| 67 | \( 1 - 2.27T + 67T^{2} \) |
| 71 | \( 1 + 0.782T + 71T^{2} \) |
| 73 | \( 1 + 1.86T + 73T^{2} \) |
| 79 | \( 1 + 8.08T + 79T^{2} \) |
| 83 | \( 1 + 15.8T + 83T^{2} \) |
| 89 | \( 1 + 11.8T + 89T^{2} \) |
| 97 | \( 1 + 15.0T + 97T^{2} \) |
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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
−8.085822830331745043823206951676, −7.59538383752524923666113944645, −7.16490903446515738027383979191, −5.62350141335110863420245195331, −5.25409019838056900280280519343, −4.20672799512062730155830827841, −3.11421463175290000119842797568, −2.44020690715435400912951400490, −1.48455686474206213660930715879, 0,
1.48455686474206213660930715879, 2.44020690715435400912951400490, 3.11421463175290000119842797568, 4.20672799512062730155830827841, 5.25409019838056900280280519343, 5.62350141335110863420245195331, 7.16490903446515738027383979191, 7.59538383752524923666113944645, 8.085822830331745043823206951676
