| L(s) = 1 | − 3·2-s − 10·3-s + 4-s + 30·6-s + 22·7-s + 21·8-s + 73·9-s − 30·11-s − 10·12-s + 46·13-s − 66·14-s − 71·16-s − 17·17-s − 219·18-s + 104·19-s − 220·21-s + 90·22-s − 42·23-s − 210·24-s − 138·26-s − 460·27-s + 22·28-s − 66·29-s + 194·31-s + 45·32-s + 300·33-s + 51·34-s + ⋯ |
| L(s) = 1 | − 1.06·2-s − 1.92·3-s + 1/8·4-s + 2.04·6-s + 1.18·7-s + 0.928·8-s + 2.70·9-s − 0.822·11-s − 0.240·12-s + 0.981·13-s − 1.25·14-s − 1.10·16-s − 0.242·17-s − 2.86·18-s + 1.25·19-s − 2.28·21-s + 0.872·22-s − 0.380·23-s − 1.78·24-s − 1.04·26-s − 3.27·27-s + 0.148·28-s − 0.422·29-s + 1.12·31-s + 0.248·32-s + 1.58·33-s + 0.257·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.5486202408\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5486202408\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 17 | \( 1 + p T \) |
| good | 2 | \( 1 + 3 T + p^{3} T^{2} \) |
| 3 | \( 1 + 10 T + p^{3} T^{2} \) |
| 7 | \( 1 - 22 T + p^{3} T^{2} \) |
| 11 | \( 1 + 30 T + p^{3} T^{2} \) |
| 13 | \( 1 - 46 T + p^{3} T^{2} \) |
| 19 | \( 1 - 104 T + p^{3} T^{2} \) |
| 23 | \( 1 + 42 T + p^{3} T^{2} \) |
| 29 | \( 1 + 66 T + p^{3} T^{2} \) |
| 31 | \( 1 - 194 T + p^{3} T^{2} \) |
| 37 | \( 1 + 206 T + p^{3} T^{2} \) |
| 41 | \( 1 + 126 T + p^{3} T^{2} \) |
| 43 | \( 1 - 388 T + p^{3} T^{2} \) |
| 47 | \( 1 - 540 T + p^{3} T^{2} \) |
| 53 | \( 1 + 78 T + p^{3} T^{2} \) |
| 59 | \( 1 - 432 T + p^{3} T^{2} \) |
| 61 | \( 1 + 10 p T + p^{3} T^{2} \) |
| 67 | \( 1 + 848 T + p^{3} T^{2} \) |
| 71 | \( 1 + 174 T + p^{3} T^{2} \) |
| 73 | \( 1 + 362 T + p^{3} T^{2} \) |
| 79 | \( 1 - 398 T + p^{3} T^{2} \) |
| 83 | \( 1 + 828 T + p^{3} T^{2} \) |
| 89 | \( 1 - 630 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1486 T + p^{3} T^{2} \) |
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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
−10.66196884350180345774591231256, −10.23016163147143113071663906037, −9.028639597838083545301883241853, −7.87793420481590915679490498482, −7.23710843871314645099202254728, −5.92506316892994558964087507450, −5.10982805750741240152202577965, −4.28915911586348537887838087764, −1.58530705597741476523497768234, −0.67482304645350152567958987713,
0.67482304645350152567958987713, 1.58530705597741476523497768234, 4.28915911586348537887838087764, 5.10982805750741240152202577965, 5.92506316892994558964087507450, 7.23710843871314645099202254728, 7.87793420481590915679490498482, 9.028639597838083545301883241853, 10.23016163147143113071663906037, 10.66196884350180345774591231256
