| L(s) = 1 | − 2.54·2-s − 2.14·3-s + 4.48·4-s − 1.97·5-s + 5.46·6-s + 3.56·7-s − 6.32·8-s + 1.60·9-s + 5.02·10-s − 1.79·11-s − 9.62·12-s − 9.07·14-s + 4.23·15-s + 7.14·16-s − 2.37·17-s − 4.09·18-s + 6.43·19-s − 8.85·20-s − 7.65·21-s + 4.57·22-s + 3.16·23-s + 13.5·24-s − 1.10·25-s + 2.98·27-s + 15.9·28-s − 8.54·29-s − 10.7·30-s + ⋯ |
| L(s) = 1 | − 1.80·2-s − 1.23·3-s + 2.24·4-s − 0.882·5-s + 2.23·6-s + 1.34·7-s − 2.23·8-s + 0.536·9-s + 1.58·10-s − 0.542·11-s − 2.77·12-s − 2.42·14-s + 1.09·15-s + 1.78·16-s − 0.575·17-s − 0.965·18-s + 1.47·19-s − 1.97·20-s − 1.66·21-s + 0.976·22-s + 0.660·23-s + 2.77·24-s − 0.220·25-s + 0.575·27-s + 3.02·28-s − 1.58·29-s − 1.96·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5239 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5239 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 \) |
| 31 | \( 1 + T \) |
| good | 2 | \( 1 + 2.54T + 2T^{2} \) |
| 3 | \( 1 + 2.14T + 3T^{2} \) |
| 5 | \( 1 + 1.97T + 5T^{2} \) |
| 7 | \( 1 - 3.56T + 7T^{2} \) |
| 11 | \( 1 + 1.79T + 11T^{2} \) |
| 17 | \( 1 + 2.37T + 17T^{2} \) |
| 19 | \( 1 - 6.43T + 19T^{2} \) |
| 23 | \( 1 - 3.16T + 23T^{2} \) |
| 29 | \( 1 + 8.54T + 29T^{2} \) |
| 37 | \( 1 - 7.43T + 37T^{2} \) |
| 41 | \( 1 - 6.15T + 41T^{2} \) |
| 43 | \( 1 + 9.24T + 43T^{2} \) |
| 47 | \( 1 + 5.38T + 47T^{2} \) |
| 53 | \( 1 - 4.02T + 53T^{2} \) |
| 59 | \( 1 + 3.90T + 59T^{2} \) |
| 61 | \( 1 - 9.88T + 61T^{2} \) |
| 67 | \( 1 + 11.7T + 67T^{2} \) |
| 71 | \( 1 - 3.78T + 71T^{2} \) |
| 73 | \( 1 - 5.31T + 73T^{2} \) |
| 79 | \( 1 + 3.77T + 79T^{2} \) |
| 83 | \( 1 - 5.34T + 83T^{2} \) |
| 89 | \( 1 + 14.7T + 89T^{2} \) |
| 97 | \( 1 + 4.79T + 97T^{2} \) |
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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
−7.893097957261660770293832030412, −7.38605763825856929696099390018, −6.77480557717948529243209961334, −5.72632130251237986612187772145, −5.17564457294781603025054761523, −4.26723229334160440591467328044, −2.96645250210339223269202974546, −1.81735917710635432458222752474, −0.934930911748851799840340164001, 0,
0.934930911748851799840340164001, 1.81735917710635432458222752474, 2.96645250210339223269202974546, 4.26723229334160440591467328044, 5.17564457294781603025054761523, 5.72632130251237986612187772145, 6.77480557717948529243209961334, 7.38605763825856929696099390018, 7.893097957261660770293832030412
