| L(s) = 1 | − 0.402·3-s − 2.83·9-s − 5.46·11-s − 4.97·13-s − 6.78·17-s − 0.859·19-s + 5.40·23-s + 2.34·27-s − 9.29·29-s + 6.99·31-s + 2.19·33-s + 2.76·37-s + 2·39-s − 9.53·41-s − 4.76·43-s − 11.1·47-s + 2.73·51-s − 1.89·53-s + 0.345·57-s + 11.2·59-s − 5.50·61-s − 3.86·67-s − 2.17·69-s − 10.4·71-s − 0.173·73-s + 5.13·79-s + 7.56·81-s + ⋯ |
| L(s) = 1 | − 0.232·3-s − 0.946·9-s − 1.64·11-s − 1.37·13-s − 1.64·17-s − 0.197·19-s + 1.12·23-s + 0.452·27-s − 1.72·29-s + 1.25·31-s + 0.382·33-s + 0.453·37-s + 0.320·39-s − 1.48·41-s − 0.725·43-s − 1.62·47-s + 0.382·51-s − 0.259·53-s + 0.0458·57-s + 1.45·59-s − 0.704·61-s − 0.471·67-s − 0.261·69-s − 1.23·71-s − 0.0202·73-s + 0.577·79-s + 0.841·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2432205103\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2432205103\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 0.402T + 3T^{2} \) |
| 11 | \( 1 + 5.46T + 11T^{2} \) |
| 13 | \( 1 + 4.97T + 13T^{2} \) |
| 17 | \( 1 + 6.78T + 17T^{2} \) |
| 19 | \( 1 + 0.859T + 19T^{2} \) |
| 23 | \( 1 - 5.40T + 23T^{2} \) |
| 29 | \( 1 + 9.29T + 29T^{2} \) |
| 31 | \( 1 - 6.99T + 31T^{2} \) |
| 37 | \( 1 - 2.76T + 37T^{2} \) |
| 41 | \( 1 + 9.53T + 41T^{2} \) |
| 43 | \( 1 + 4.76T + 43T^{2} \) |
| 47 | \( 1 + 11.1T + 47T^{2} \) |
| 53 | \( 1 + 1.89T + 53T^{2} \) |
| 59 | \( 1 - 11.2T + 59T^{2} \) |
| 61 | \( 1 + 5.50T + 61T^{2} \) |
| 67 | \( 1 + 3.86T + 67T^{2} \) |
| 71 | \( 1 + 10.4T + 71T^{2} \) |
| 73 | \( 1 + 0.173T + 73T^{2} \) |
| 79 | \( 1 - 5.13T + 79T^{2} \) |
| 83 | \( 1 + 6.43T + 83T^{2} \) |
| 89 | \( 1 + 6.51T + 89T^{2} \) |
| 97 | \( 1 - 8.94T + 97T^{2} \) |
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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
−7.67767790532953642003161127655, −6.97067982001333865772292539399, −6.37740247356367943747969995288, −5.40660228576656697146720975414, −5.04238423221428509276275726726, −4.43782724549576037438939873855, −3.15779035140609046478807211814, −2.64603928689946519957621014965, −1.92384861008077836865408261823, −0.21701236098198348692272855107,
0.21701236098198348692272855107, 1.92384861008077836865408261823, 2.64603928689946519957621014965, 3.15779035140609046478807211814, 4.43782724549576037438939873855, 5.04238423221428509276275726726, 5.40660228576656697146720975414, 6.37740247356367943747969995288, 6.97067982001333865772292539399, 7.67767790532953642003161127655
