L(s) = 1 | + 2.76·2-s + 1.76·3-s + 5.62·4-s − 2.62·5-s + 4.86·6-s + 7-s + 10.0·8-s + 0.103·9-s − 7.25·10-s + 9.91·12-s − 2.38·13-s + 2.76·14-s − 4.62·15-s + 16.4·16-s − 2.38·17-s + 0.284·18-s − 1.72·19-s − 14.7·20-s + 1.76·21-s − 0.626·23-s + 17.6·24-s + 1.89·25-s − 6.59·26-s − 5.10·27-s + 5.62·28-s + 1.72·29-s − 12.7·30-s + ⋯ |
L(s) = 1 | + 1.95·2-s + 1.01·3-s + 2.81·4-s − 1.17·5-s + 1.98·6-s + 0.377·7-s + 3.54·8-s + 0.0343·9-s − 2.29·10-s + 2.86·12-s − 0.662·13-s + 0.738·14-s − 1.19·15-s + 4.10·16-s − 0.579·17-s + 0.0670·18-s − 0.396·19-s − 3.30·20-s + 0.384·21-s − 0.130·23-s + 3.60·24-s + 0.379·25-s − 1.29·26-s − 0.982·27-s + 1.06·28-s + 0.321·29-s − 2.33·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.776814555\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.776814555\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 - 2.76T + 2T^{2} \) |
| 3 | \( 1 - 1.76T + 3T^{2} \) |
| 5 | \( 1 + 2.62T + 5T^{2} \) |
| 13 | \( 1 + 2.38T + 13T^{2} \) |
| 17 | \( 1 + 2.38T + 17T^{2} \) |
| 19 | \( 1 + 1.72T + 19T^{2} \) |
| 23 | \( 1 + 0.626T + 23T^{2} \) |
| 29 | \( 1 - 1.72T + 29T^{2} \) |
| 31 | \( 1 + 2.23T + 31T^{2} \) |
| 37 | \( 1 + 6.89T + 37T^{2} \) |
| 41 | \( 1 - 10.3T + 41T^{2} \) |
| 43 | \( 1 - 7.25T + 43T^{2} \) |
| 47 | \( 1 - 6.38T + 47T^{2} \) |
| 53 | \( 1 + 9.25T + 53T^{2} \) |
| 59 | \( 1 + 1.76T + 59T^{2} \) |
| 61 | \( 1 + 10.3T + 61T^{2} \) |
| 67 | \( 1 + 6.42T + 67T^{2} \) |
| 71 | \( 1 - 8.08T + 71T^{2} \) |
| 73 | \( 1 - 10.3T + 73T^{2} \) |
| 79 | \( 1 - 15.2T + 79T^{2} \) |
| 83 | \( 1 - 12.7T + 83T^{2} \) |
| 89 | \( 1 + 14.1T + 89T^{2} \) |
| 97 | \( 1 + 8.35T + 97T^{2} \) |
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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
−10.78093803617624314488667484020, −9.243809070051289830383274652668, −8.002349528625040366798296272596, −7.61320147668350581797608037431, −6.66611460654339343166775326581, −5.53312322763378273640079472407, −4.48622256257557296914836699792, −3.91151475455870814715786221651, −2.97939239094261403005347015760, −2.11164309607064355327692554514,
2.11164309607064355327692554514, 2.97939239094261403005347015760, 3.91151475455870814715786221651, 4.48622256257557296914836699792, 5.53312322763378273640079472407, 6.66611460654339343166775326581, 7.61320147668350581797608037431, 8.002349528625040366798296272596, 9.243809070051289830383274652668, 10.78093803617624314488667484020