| L(s) = 1 | − 1.91·2-s + 1.67·3-s + 1.66·4-s + 5-s − 3.20·6-s − 4.22·7-s + 0.634·8-s − 0.204·9-s − 1.91·10-s + 2.92·11-s + 2.78·12-s − 2.06·13-s + 8.09·14-s + 1.67·15-s − 4.55·16-s + 3.59·17-s + 0.392·18-s + 1.66·20-s − 7.06·21-s − 5.60·22-s + 8.02·23-s + 1.06·24-s + 25-s + 3.95·26-s − 5.35·27-s − 7.05·28-s − 2.00·29-s + ⋯ |
| L(s) = 1 | − 1.35·2-s + 0.965·3-s + 0.834·4-s + 0.447·5-s − 1.30·6-s − 1.59·7-s + 0.224·8-s − 0.0682·9-s − 0.605·10-s + 0.882·11-s + 0.805·12-s − 0.572·13-s + 2.16·14-s + 0.431·15-s − 1.13·16-s + 0.872·17-s + 0.0924·18-s + 0.373·20-s − 1.54·21-s − 1.19·22-s + 1.67·23-s + 0.216·24-s + 0.200·25-s + 0.775·26-s − 1.03·27-s − 1.33·28-s − 0.372·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.012253731\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.012253731\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + 1.91T + 2T^{2} \) |
| 3 | \( 1 - 1.67T + 3T^{2} \) |
| 7 | \( 1 + 4.22T + 7T^{2} \) |
| 11 | \( 1 - 2.92T + 11T^{2} \) |
| 13 | \( 1 + 2.06T + 13T^{2} \) |
| 17 | \( 1 - 3.59T + 17T^{2} \) |
| 23 | \( 1 - 8.02T + 23T^{2} \) |
| 29 | \( 1 + 2.00T + 29T^{2} \) |
| 31 | \( 1 + 8.46T + 31T^{2} \) |
| 37 | \( 1 + 2.20T + 37T^{2} \) |
| 41 | \( 1 - 9.76T + 41T^{2} \) |
| 43 | \( 1 - 10.5T + 43T^{2} \) |
| 47 | \( 1 - 3.58T + 47T^{2} \) |
| 53 | \( 1 - 3.24T + 53T^{2} \) |
| 59 | \( 1 + 2.08T + 59T^{2} \) |
| 61 | \( 1 + 4.93T + 61T^{2} \) |
| 67 | \( 1 - 10.1T + 67T^{2} \) |
| 71 | \( 1 + 2.85T + 71T^{2} \) |
| 73 | \( 1 - 1.82T + 73T^{2} \) |
| 79 | \( 1 - 12.7T + 79T^{2} \) |
| 83 | \( 1 - 17.2T + 83T^{2} \) |
| 89 | \( 1 + 2.46T + 89T^{2} \) |
| 97 | \( 1 - 18.0T + 97T^{2} \) |
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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
−9.196221095016520164566696887387, −9.012953154018800946572841007127, −7.74719234608230993458586079728, −7.23442554737819071535742003136, −6.41205715161499321506287933617, −5.42586495296934550476033064461, −3.91586295408097486120073705242, −3.07032198797341454557828862010, −2.18044934582593107071487804451, −0.797043799253178322469111496397,
0.797043799253178322469111496397, 2.18044934582593107071487804451, 3.07032198797341454557828862010, 3.91586295408097486120073705242, 5.42586495296934550476033064461, 6.41205715161499321506287933617, 7.23442554737819071535742003136, 7.74719234608230993458586079728, 9.012953154018800946572841007127, 9.196221095016520164566696887387
