| L(s) = 1 | − 4.53·2-s − 6.46·3-s + 12.5·4-s + 29.3·6-s − 7·7-s − 20.7·8-s + 14.8·9-s − 54.0·11-s − 81.2·12-s − 75.2·13-s + 31.7·14-s − 6.60·16-s + 71.2·17-s − 67.1·18-s − 65.5·19-s + 45.2·21-s + 245.·22-s − 125.·23-s + 133.·24-s + 341.·26-s + 78.8·27-s − 87.9·28-s + 190.·29-s − 193.·31-s + 195.·32-s + 349.·33-s − 323.·34-s + ⋯ |
| L(s) = 1 | − 1.60·2-s − 1.24·3-s + 1.57·4-s + 1.99·6-s − 0.377·7-s − 0.915·8-s + 0.548·9-s − 1.48·11-s − 1.95·12-s − 1.60·13-s + 0.606·14-s − 0.103·16-s + 1.01·17-s − 0.879·18-s − 0.791·19-s + 0.470·21-s + 2.37·22-s − 1.13·23-s + 1.13·24-s + 2.57·26-s + 0.561·27-s − 0.593·28-s + 1.21·29-s − 1.11·31-s + 1.08·32-s + 1.84·33-s − 1.62·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.1786143992\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1786143992\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| good | 2 | \( 1 + 4.53T + 8T^{2} \) |
| 3 | \( 1 + 6.46T + 27T^{2} \) |
| 11 | \( 1 + 54.0T + 1.33e3T^{2} \) |
| 13 | \( 1 + 75.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 71.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 65.5T + 6.85e3T^{2} \) |
| 23 | \( 1 + 125.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 190.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 193.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 114.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 216.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 413.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 113.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 584.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 203.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 162.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 477.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 822.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 798.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 468.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 310.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.31e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.31e3T + 9.12e5T^{2} \) |
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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
−12.05721915596762340547820366333, −10.78951683301677361574757102354, −10.33440152613894276378283363696, −9.489585506418059313684330542014, −8.063514353850715638176607023040, −7.31327697550779981426658038184, −6.07923502038472262505001794579, −4.93650334657927347687269860537, −2.42601663334762778329942701282, −0.41960715885990573395412704509,
0.41960715885990573395412704509, 2.42601663334762778329942701282, 4.93650334657927347687269860537, 6.07923502038472262505001794579, 7.31327697550779981426658038184, 8.063514353850715638176607023040, 9.489585506418059313684330542014, 10.33440152613894276378283363696, 10.78951683301677361574757102354, 12.05721915596762340547820366333
