L(s) = 1 | − 0.414·2-s − 3-s − 1.82·4-s − 5-s + 0.414·6-s + 0.585·7-s + 1.58·8-s + 9-s + 0.414·10-s + 1.41·11-s + 1.82·12-s + 5.41·13-s − 0.242·14-s + 15-s + 3·16-s − 1.17·17-s − 0.414·18-s + 19-s + 1.82·20-s − 0.585·21-s − 0.585·22-s + 7.65·23-s − 1.58·24-s + 25-s − 2.24·26-s − 27-s − 1.07·28-s + ⋯ |
L(s) = 1 | − 0.292·2-s − 0.577·3-s − 0.914·4-s − 0.447·5-s + 0.169·6-s + 0.221·7-s + 0.560·8-s + 0.333·9-s + 0.130·10-s + 0.426·11-s + 0.527·12-s + 1.50·13-s − 0.0648·14-s + 0.258·15-s + 0.750·16-s − 0.284·17-s − 0.0976·18-s + 0.229·19-s + 0.408·20-s − 0.127·21-s − 0.124·22-s + 1.59·23-s − 0.323·24-s + 0.200·25-s − 0.439·26-s − 0.192·27-s − 0.202·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7676795962\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7676795962\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 19 | \( 1 - T \) |
good | 2 | \( 1 + 0.414T + 2T^{2} \) |
| 7 | \( 1 - 0.585T + 7T^{2} \) |
| 11 | \( 1 - 1.41T + 11T^{2} \) |
| 13 | \( 1 - 5.41T + 13T^{2} \) |
| 17 | \( 1 + 1.17T + 17T^{2} \) |
| 23 | \( 1 - 7.65T + 23T^{2} \) |
| 29 | \( 1 + 9.07T + 29T^{2} \) |
| 31 | \( 1 - 6.48T + 31T^{2} \) |
| 37 | \( 1 - 11.0T + 37T^{2} \) |
| 41 | \( 1 + 7.41T + 41T^{2} \) |
| 43 | \( 1 - 0.585T + 43T^{2} \) |
| 47 | \( 1 - 0.343T + 47T^{2} \) |
| 53 | \( 1 - 4T + 53T^{2} \) |
| 59 | \( 1 - 8.48T + 59T^{2} \) |
| 61 | \( 1 - 5.65T + 61T^{2} \) |
| 67 | \( 1 - 12T + 67T^{2} \) |
| 71 | \( 1 + 4.48T + 71T^{2} \) |
| 73 | \( 1 + 2T + 73T^{2} \) |
| 79 | \( 1 + 11.3T + 79T^{2} \) |
| 83 | \( 1 + 10.4T + 83T^{2} \) |
| 89 | \( 1 - 10.7T + 89T^{2} \) |
| 97 | \( 1 + 4.24T + 97T^{2} \) |
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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
−11.52499868984857256751197083355, −11.07373818285000871114656988342, −9.895306648110656728687637429620, −8.927391779320616342134129825796, −8.165941696646063731713821037254, −6.96848280434509019490203487705, −5.74205691389279477949312162713, −4.61747116333653192293579041578, −3.61823027123933242237126070482, −1.06457276340249043064631230540,
1.06457276340249043064631230540, 3.61823027123933242237126070482, 4.61747116333653192293579041578, 5.74205691389279477949312162713, 6.96848280434509019490203487705, 8.165941696646063731713821037254, 8.927391779320616342134129825796, 9.895306648110656728687637429620, 11.07373818285000871114656988342, 11.52499868984857256751197083355