| L(s) = 1 | + 3-s − 1.37·5-s + 3.37·7-s + 9-s + 1.37·11-s + 2·13-s − 1.37·15-s + 1.37·17-s + 19-s + 3.37·21-s − 8.74·23-s − 3.11·25-s + 27-s + 2.74·29-s − 6.74·31-s + 1.37·33-s − 4.62·35-s + 4.74·37-s + 2·39-s + 3.37·43-s − 1.37·45-s − 13.3·47-s + 4.37·49-s + 1.37·51-s − 2.74·53-s − 1.88·55-s + 57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.613·5-s + 1.27·7-s + 0.333·9-s + 0.413·11-s + 0.554·13-s − 0.354·15-s + 0.332·17-s + 0.229·19-s + 0.735·21-s − 1.82·23-s − 0.623·25-s + 0.192·27-s + 0.509·29-s − 1.21·31-s + 0.238·33-s − 0.782·35-s + 0.780·37-s + 0.320·39-s + 0.514·43-s − 0.204·45-s − 1.95·47-s + 0.624·49-s + 0.192·51-s − 0.376·53-s − 0.253·55-s + 0.132·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 228 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 228 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.503340340\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.503340340\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 + 1.37T + 5T^{2} \) |
| 7 | \( 1 - 3.37T + 7T^{2} \) |
| 11 | \( 1 - 1.37T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 - 1.37T + 17T^{2} \) |
| 23 | \( 1 + 8.74T + 23T^{2} \) |
| 29 | \( 1 - 2.74T + 29T^{2} \) |
| 31 | \( 1 + 6.74T + 31T^{2} \) |
| 37 | \( 1 - 4.74T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 3.37T + 43T^{2} \) |
| 47 | \( 1 + 13.3T + 47T^{2} \) |
| 53 | \( 1 + 2.74T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 2.62T + 61T^{2} \) |
| 67 | \( 1 + 9.48T + 67T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 + 5.37T + 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 - 8.74T + 83T^{2} \) |
| 89 | \( 1 - 14.7T + 89T^{2} \) |
| 97 | \( 1 - 14T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.99909326968577604765843803698, −11.43621180026715616611070099735, −10.32879129388057254639559594925, −9.141584942465485639443873380424, −8.083983779024197901234437123539, −7.63888827472407111571152699129, −6.07002567274795462214650090870, −4.61343299238769059810422748930, −3.60641570226842431096390878579, −1.76610847723928385600833153251,
1.76610847723928385600833153251, 3.60641570226842431096390878579, 4.61343299238769059810422748930, 6.07002567274795462214650090870, 7.63888827472407111571152699129, 8.083983779024197901234437123539, 9.141584942465485639443873380424, 10.32879129388057254639559594925, 11.43621180026715616611070099735, 11.99909326968577604765843803698
