| L(s) = 1 | + 2.40·2-s + 3-s + 3.77·4-s − 5-s + 2.40·6-s − 7-s + 4.25·8-s + 9-s − 2.40·10-s + 2.12·11-s + 3.77·12-s + 2.58·13-s − 2.40·14-s − 15-s + 2.67·16-s + 2.79·17-s + 2.40·18-s + 6.41·19-s − 3.77·20-s − 21-s + 5.10·22-s − 23-s + 4.25·24-s + 25-s + 6.19·26-s + 27-s − 3.77·28-s + ⋯ |
| L(s) = 1 | + 1.69·2-s + 0.577·3-s + 1.88·4-s − 0.447·5-s + 0.980·6-s − 0.377·7-s + 1.50·8-s + 0.333·9-s − 0.759·10-s + 0.641·11-s + 1.08·12-s + 0.715·13-s − 0.642·14-s − 0.258·15-s + 0.669·16-s + 0.678·17-s + 0.566·18-s + 1.47·19-s − 0.843·20-s − 0.218·21-s + 1.08·22-s − 0.208·23-s + 0.868·24-s + 0.200·25-s + 1.21·26-s + 0.192·27-s − 0.712·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2415 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2415 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.986252935\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.986252935\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 - 2.40T + 2T^{2} \) |
| 11 | \( 1 - 2.12T + 11T^{2} \) |
| 13 | \( 1 - 2.58T + 13T^{2} \) |
| 17 | \( 1 - 2.79T + 17T^{2} \) |
| 19 | \( 1 - 6.41T + 19T^{2} \) |
| 29 | \( 1 + 5.89T + 29T^{2} \) |
| 31 | \( 1 + 4.23T + 31T^{2} \) |
| 37 | \( 1 - 9.81T + 37T^{2} \) |
| 41 | \( 1 - 11.8T + 41T^{2} \) |
| 43 | \( 1 - 3.96T + 43T^{2} \) |
| 47 | \( 1 + 8.94T + 47T^{2} \) |
| 53 | \( 1 + 7.63T + 53T^{2} \) |
| 59 | \( 1 - 0.422T + 59T^{2} \) |
| 61 | \( 1 + 6.69T + 61T^{2} \) |
| 67 | \( 1 + 3.05T + 67T^{2} \) |
| 71 | \( 1 + 11.4T + 71T^{2} \) |
| 73 | \( 1 + 3.17T + 73T^{2} \) |
| 79 | \( 1 + 2.35T + 79T^{2} \) |
| 83 | \( 1 - 14.1T + 83T^{2} \) |
| 89 | \( 1 - 8.92T + 89T^{2} \) |
| 97 | \( 1 - 0.838T + 97T^{2} \) |
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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
−9.100706907068198323002633922105, −7.73089420452454318995256178164, −7.44889498807603345535083521358, −6.31255039160014005293225064404, −5.82689348513290386846806531393, −4.83730987989739114078625017486, −3.93301714375348295839685708105, −3.47216845724950046618912465725, −2.70018111238315871101269870032, −1.36478370188527073896342591071,
1.36478370188527073896342591071, 2.70018111238315871101269870032, 3.47216845724950046618912465725, 3.93301714375348295839685708105, 4.83730987989739114078625017486, 5.82689348513290386846806531393, 6.31255039160014005293225064404, 7.44889498807603345535083521358, 7.73089420452454318995256178164, 9.100706907068198323002633922105
