| L(s) = 1 | + 1.11·2-s − 2.85·3-s − 0.757·4-s + 3.45·5-s − 3.18·6-s + 7-s − 3.07·8-s + 5.17·9-s + 3.85·10-s + 2.16·12-s − 2.05·13-s + 1.11·14-s − 9.88·15-s − 1.90·16-s + 1.93·17-s + 5.76·18-s + 1.62·19-s − 2.61·20-s − 2.85·21-s − 0.807·23-s + 8.78·24-s + 6.94·25-s − 2.29·26-s − 6.21·27-s − 0.757·28-s + 7.97·29-s − 11.0·30-s + ⋯ |
| L(s) = 1 | + 0.788·2-s − 1.65·3-s − 0.378·4-s + 1.54·5-s − 1.30·6-s + 0.377·7-s − 1.08·8-s + 1.72·9-s + 1.21·10-s + 0.625·12-s − 0.571·13-s + 0.297·14-s − 2.55·15-s − 0.477·16-s + 0.468·17-s + 1.35·18-s + 0.372·19-s − 0.585·20-s − 0.623·21-s − 0.168·23-s + 1.79·24-s + 1.38·25-s − 0.450·26-s − 1.19·27-s − 0.143·28-s + 1.48·29-s − 2.01·30-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
| \(L(1)\) |
\(\approx\) |
\(1.513630185\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.513630185\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.11T + 2T^{2} \) |
| 3 | \( 1 + 2.85T + 3T^{2} \) |
| 5 | \( 1 - 3.45T + 5T^{2} \) |
| 13 | \( 1 + 2.05T + 13T^{2} \) |
| 17 | \( 1 - 1.93T + 17T^{2} \) |
| 19 | \( 1 - 1.62T + 19T^{2} \) |
| 23 | \( 1 + 0.807T + 23T^{2} \) |
| 29 | \( 1 - 7.97T + 29T^{2} \) |
| 31 | \( 1 - 0.788T + 31T^{2} \) |
| 37 | \( 1 - 10.0T + 37T^{2} \) |
| 41 | \( 1 + 2.12T + 41T^{2} \) |
| 43 | \( 1 - 3.08T + 43T^{2} \) |
| 47 | \( 1 - 7.56T + 47T^{2} \) |
| 53 | \( 1 - 10.8T + 53T^{2} \) |
| 59 | \( 1 + 3.29T + 59T^{2} \) |
| 61 | \( 1 + 1.07T + 61T^{2} \) |
| 67 | \( 1 - 2.40T + 67T^{2} \) |
| 71 | \( 1 + 3.18T + 71T^{2} \) |
| 73 | \( 1 - 1.22T + 73T^{2} \) |
| 79 | \( 1 + 9.48T + 79T^{2} \) |
| 83 | \( 1 + 16.0T + 83T^{2} \) |
| 89 | \( 1 + 4.43T + 89T^{2} \) |
| 97 | \( 1 - 6.46T + 97T^{2} \) |
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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
−10.09655390754372296660943892881, −9.796893040437357520230162676636, −8.667626572514585301642434947324, −7.19985395165180400224200808933, −6.14122909021800143591540661594, −5.76594885500718593266025218149, −5.04028840049940463549099705412, −4.35516531437021935461518048227, −2.62659087405395493953217753664, −1.01459601045708145917534886069,
1.01459601045708145917534886069, 2.62659087405395493953217753664, 4.35516531437021935461518048227, 5.04028840049940463549099705412, 5.76594885500718593266025218149, 6.14122909021800143591540661594, 7.19985395165180400224200808933, 8.667626572514585301642434947324, 9.796893040437357520230162676636, 10.09655390754372296660943892881
