| L(s) = 1 | − 2.36·2-s + 3-s + 3.59·4-s + 5-s − 2.36·6-s + 0.0275·7-s − 3.77·8-s + 9-s − 2.36·10-s + 5.24·11-s + 3.59·12-s + 5.51·13-s − 0.0652·14-s + 15-s + 1.73·16-s + 1.18·17-s − 2.36·18-s − 2.14·19-s + 3.59·20-s + 0.0275·21-s − 12.3·22-s − 3.77·24-s + 25-s − 13.0·26-s + 27-s + 0.0991·28-s − 9.85·29-s + ⋯ |
| L(s) = 1 | − 1.67·2-s + 0.577·3-s + 1.79·4-s + 0.447·5-s − 0.965·6-s + 0.0104·7-s − 1.33·8-s + 0.333·9-s − 0.747·10-s + 1.58·11-s + 1.03·12-s + 1.52·13-s − 0.0174·14-s + 0.258·15-s + 0.432·16-s + 0.287·17-s − 0.557·18-s − 0.492·19-s + 0.803·20-s + 0.00602·21-s − 2.64·22-s − 0.769·24-s + 0.200·25-s − 2.55·26-s + 0.192·27-s + 0.0187·28-s − 1.82·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.558330531\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.558330531\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 2 | \( 1 + 2.36T + 2T^{2} \) |
| 7 | \( 1 - 0.0275T + 7T^{2} \) |
| 11 | \( 1 - 5.24T + 11T^{2} \) |
| 13 | \( 1 - 5.51T + 13T^{2} \) |
| 17 | \( 1 - 1.18T + 17T^{2} \) |
| 19 | \( 1 + 2.14T + 19T^{2} \) |
| 29 | \( 1 + 9.85T + 29T^{2} \) |
| 31 | \( 1 - 2.87T + 31T^{2} \) |
| 37 | \( 1 + 9.17T + 37T^{2} \) |
| 41 | \( 1 + 4.10T + 41T^{2} \) |
| 43 | \( 1 + 1.11T + 43T^{2} \) |
| 47 | \( 1 - 0.209T + 47T^{2} \) |
| 53 | \( 1 - 2.85T + 53T^{2} \) |
| 59 | \( 1 - 12.1T + 59T^{2} \) |
| 61 | \( 1 - 9.75T + 61T^{2} \) |
| 67 | \( 1 - 8.19T + 67T^{2} \) |
| 71 | \( 1 - 16.1T + 71T^{2} \) |
| 73 | \( 1 + 3.73T + 73T^{2} \) |
| 79 | \( 1 - 2.17T + 79T^{2} \) |
| 83 | \( 1 - 10.3T + 83T^{2} \) |
| 89 | \( 1 - 3.00T + 89T^{2} \) |
| 97 | \( 1 + 2.52T + 97T^{2} \) |
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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
−8.194783234330227843135173053932, −7.26682046353261628355159353178, −6.64347409607241381540080859211, −6.23990592566707544494300645089, −5.20728699769315123375722987646, −3.86647349563074852282518848409, −3.51145083734358664787062752376, −2.14274710363131191160257924661, −1.62985615222501610941152857782, −0.832694785638239461388079267892,
0.832694785638239461388079267892, 1.62985615222501610941152857782, 2.14274710363131191160257924661, 3.51145083734358664787062752376, 3.86647349563074852282518848409, 5.20728699769315123375722987646, 6.23990592566707544494300645089, 6.64347409607241381540080859211, 7.26682046353261628355159353178, 8.194783234330227843135173053932
