| L(s) = 1 | − 2-s + 3·3-s + 4-s − 3·6-s − 8-s + 6·9-s − 11-s + 3·12-s + 2·13-s + 16-s + 7·17-s − 6·18-s + 5·19-s + 22-s − 6·23-s − 3·24-s − 2·26-s + 9·27-s − 4·31-s − 32-s − 3·33-s − 7·34-s + 6·36-s − 37-s − 5·38-s + 6·39-s − 3·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.73·3-s + 1/2·4-s − 1.22·6-s − 0.353·8-s + 2·9-s − 0.301·11-s + 0.866·12-s + 0.554·13-s + 1/4·16-s + 1.69·17-s − 1.41·18-s + 1.14·19-s + 0.213·22-s − 1.25·23-s − 0.612·24-s − 0.392·26-s + 1.73·27-s − 0.718·31-s − 0.176·32-s − 0.522·33-s − 1.20·34-s + 36-s − 0.164·37-s − 0.811·38-s + 0.960·39-s − 0.468·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.558578453\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.558578453\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 3 | \( 1 - p T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 + 7 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−9.177649149000980779645265803026, −8.459325576811713568506936791912, −7.68736201597947535128658618989, −7.53300371838530950412027477632, −6.25262510775209233600867082738, −5.20732494807423981373300369721, −3.74965680760340825728557237450, −3.27661932383606895541022078526, −2.23400282973532377904024408291, −1.23504527578568251780025983879,
1.23504527578568251780025983879, 2.23400282973532377904024408291, 3.27661932383606895541022078526, 3.74965680760340825728557237450, 5.20732494807423981373300369721, 6.25262510775209233600867082738, 7.53300371838530950412027477632, 7.68736201597947535128658618989, 8.459325576811713568506936791912, 9.177649149000980779645265803026
