| L(s) = 1 | + 1.37·3-s − 3.70·5-s − 0.849·7-s − 1.09·9-s + 4.69·11-s + 1.63·13-s − 5.11·15-s − 5.80·17-s + 5.63·19-s − 1.17·21-s − 0.392·23-s + 8.73·25-s − 5.65·27-s + 29-s + 6.74·31-s + 6.47·33-s + 3.15·35-s + 3.56·37-s + 2.25·39-s + 0.0925·41-s + 4.69·43-s + 4.07·45-s − 2.45·47-s − 6.27·49-s − 8.00·51-s + 2.36·53-s − 17.4·55-s + ⋯ |
| L(s) = 1 | + 0.795·3-s − 1.65·5-s − 0.321·7-s − 0.366·9-s + 1.41·11-s + 0.452·13-s − 1.31·15-s − 1.40·17-s + 1.29·19-s − 0.255·21-s − 0.0818·23-s + 1.74·25-s − 1.08·27-s + 0.185·29-s + 1.21·31-s + 1.12·33-s + 0.532·35-s + 0.585·37-s + 0.360·39-s + 0.0144·41-s + 0.716·43-s + 0.607·45-s − 0.358·47-s − 0.896·49-s − 1.12·51-s + 0.325·53-s − 2.34·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.581704951\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.581704951\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 - 1.37T + 3T^{2} \) |
| 5 | \( 1 + 3.70T + 5T^{2} \) |
| 7 | \( 1 + 0.849T + 7T^{2} \) |
| 11 | \( 1 - 4.69T + 11T^{2} \) |
| 13 | \( 1 - 1.63T + 13T^{2} \) |
| 17 | \( 1 + 5.80T + 17T^{2} \) |
| 19 | \( 1 - 5.63T + 19T^{2} \) |
| 23 | \( 1 + 0.392T + 23T^{2} \) |
| 31 | \( 1 - 6.74T + 31T^{2} \) |
| 37 | \( 1 - 3.56T + 37T^{2} \) |
| 41 | \( 1 - 0.0925T + 41T^{2} \) |
| 43 | \( 1 - 4.69T + 43T^{2} \) |
| 47 | \( 1 + 2.45T + 47T^{2} \) |
| 53 | \( 1 - 2.36T + 53T^{2} \) |
| 59 | \( 1 - 2.09T + 59T^{2} \) |
| 61 | \( 1 - 13.0T + 61T^{2} \) |
| 67 | \( 1 - 8.97T + 67T^{2} \) |
| 71 | \( 1 - 16.0T + 71T^{2} \) |
| 73 | \( 1 - 15.1T + 73T^{2} \) |
| 79 | \( 1 + 3.06T + 79T^{2} \) |
| 83 | \( 1 + 1.37T + 83T^{2} \) |
| 89 | \( 1 - 12.2T + 89T^{2} \) |
| 97 | \( 1 - 9.07T + 97T^{2} \) |
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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
−9.092897830945933412459691403975, −8.385387591556435887831146837286, −7.900210254556346878744348977497, −6.92852536287119118470350637151, −6.34353801670740553359690616753, −4.92907029697691122135222943782, −3.89725813736352858456316105658, −3.57955717533243372112184405972, −2.50036613316747884394700972505, −0.824066278820617063455714696182,
0.824066278820617063455714696182, 2.50036613316747884394700972505, 3.57955717533243372112184405972, 3.89725813736352858456316105658, 4.92907029697691122135222943782, 6.34353801670740553359690616753, 6.92852536287119118470350637151, 7.900210254556346878744348977497, 8.385387591556435887831146837286, 9.092897830945933412459691403975
