| L(s) = 1 | + 2.01·3-s + 3.16·5-s + 2.95·7-s + 1.04·9-s + 2.62·11-s + 5.88·13-s + 6.36·15-s − 6.15·17-s − 6.44·19-s + 5.93·21-s + 4.28·23-s + 5.01·25-s − 3.92·27-s − 4.42·29-s + 4.31·31-s + 5.28·33-s + 9.33·35-s + 7.32·37-s + 11.8·39-s − 6.77·41-s + 0.439·43-s + 3.31·45-s + 6.08·47-s + 1.70·49-s − 12.3·51-s − 10.8·53-s + 8.31·55-s + ⋯ |
| L(s) = 1 | + 1.16·3-s + 1.41·5-s + 1.11·7-s + 0.349·9-s + 0.792·11-s + 1.63·13-s + 1.64·15-s − 1.49·17-s − 1.47·19-s + 1.29·21-s + 0.892·23-s + 1.00·25-s − 0.755·27-s − 0.822·29-s + 0.774·31-s + 0.920·33-s + 1.57·35-s + 1.20·37-s + 1.89·39-s − 1.05·41-s + 0.0670·43-s + 0.494·45-s + 0.886·47-s + 0.243·49-s − 1.73·51-s − 1.48·53-s + 1.12·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.625316090\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.625316090\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 251 | \( 1 - T \) |
| good | 3 | \( 1 - 2.01T + 3T^{2} \) |
| 5 | \( 1 - 3.16T + 5T^{2} \) |
| 7 | \( 1 - 2.95T + 7T^{2} \) |
| 11 | \( 1 - 2.62T + 11T^{2} \) |
| 13 | \( 1 - 5.88T + 13T^{2} \) |
| 17 | \( 1 + 6.15T + 17T^{2} \) |
| 19 | \( 1 + 6.44T + 19T^{2} \) |
| 23 | \( 1 - 4.28T + 23T^{2} \) |
| 29 | \( 1 + 4.42T + 29T^{2} \) |
| 31 | \( 1 - 4.31T + 31T^{2} \) |
| 37 | \( 1 - 7.32T + 37T^{2} \) |
| 41 | \( 1 + 6.77T + 41T^{2} \) |
| 43 | \( 1 - 0.439T + 43T^{2} \) |
| 47 | \( 1 - 6.08T + 47T^{2} \) |
| 53 | \( 1 + 10.8T + 53T^{2} \) |
| 59 | \( 1 - 9.72T + 59T^{2} \) |
| 61 | \( 1 - 13.3T + 61T^{2} \) |
| 67 | \( 1 + 3.57T + 67T^{2} \) |
| 71 | \( 1 + 15.9T + 71T^{2} \) |
| 73 | \( 1 + 1.54T + 73T^{2} \) |
| 79 | \( 1 + 1.70T + 79T^{2} \) |
| 83 | \( 1 - 10.4T + 83T^{2} \) |
| 89 | \( 1 + 9.71T + 89T^{2} \) |
| 97 | \( 1 + 8.18T + 97T^{2} \) |
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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
−8.605819288553321149096245656030, −8.074234409139200664308448910092, −6.84599029770624524758498468149, −6.30844189451083027997035583941, −5.57347787524723374533908840605, −4.49643684117926730656593412274, −3.88123628406486548659062064894, −2.70291338356638817103646486024, −1.97644112198799484991995693326, −1.37848591839532362779755997970,
1.37848591839532362779755997970, 1.97644112198799484991995693326, 2.70291338356638817103646486024, 3.88123628406486548659062064894, 4.49643684117926730656593412274, 5.57347787524723374533908840605, 6.30844189451083027997035583941, 6.84599029770624524758498468149, 8.074234409139200664308448910092, 8.605819288553321149096245656030
