| L(s) = 1 | + 2-s − 4-s + 2·5-s + 7-s − 3·8-s + 2·10-s − 6·11-s + 13-s + 14-s − 16-s + 5·19-s − 2·20-s − 6·22-s + 2·23-s − 25-s + 26-s − 28-s + 6·29-s + 7·31-s + 5·32-s + 2·35-s + 7·37-s + 5·38-s − 6·40-s + 6·41-s + 7·43-s + 6·44-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s + 0.894·5-s + 0.377·7-s − 1.06·8-s + 0.632·10-s − 1.80·11-s + 0.277·13-s + 0.267·14-s − 1/4·16-s + 1.14·19-s − 0.447·20-s − 1.27·22-s + 0.417·23-s − 1/5·25-s + 0.196·26-s − 0.188·28-s + 1.11·29-s + 1.25·31-s + 0.883·32-s + 0.338·35-s + 1.15·37-s + 0.811·38-s − 0.948·40-s + 0.937·41-s + 1.06·43-s + 0.904·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.434823122\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.434823122\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 11 T + p T^{2} \) |
| 67 | \( 1 + 11 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 + 16 T + p T^{2} \) |
| 97 | \( 1 - 17 T + p T^{2} \) |
| 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
−8.886315830436409117872754341444, −8.100168941700196578993883102653, −7.44250795054134353693959563274, −6.15697273398811664409499252389, −5.68002927616461946621638276962, −4.97715800039479722460225575345, −4.34712382606325490933530950641, −3.02164880973273933564814222892, −2.49135863560126445232028743601, −0.899888649100903198174558343945,
0.899888649100903198174558343945, 2.49135863560126445232028743601, 3.02164880973273933564814222892, 4.34712382606325490933530950641, 4.97715800039479722460225575345, 5.68002927616461946621638276962, 6.15697273398811664409499252389, 7.44250795054134353693959563274, 8.100168941700196578993883102653, 8.886315830436409117872754341444
