| L(s) = 1 | + 2.30·2-s − 2.30·3-s + 3.30·4-s + 3.60·5-s − 5.30·6-s − 7-s + 3.00·8-s + 2.30·9-s + 8.30·10-s − 7.60·12-s + 6.60·13-s − 2.30·14-s − 8.30·15-s + 0.302·16-s + 2.69·17-s + 5.30·18-s − 3·19-s + 11.9·20-s + 2.30·21-s − 2.69·23-s − 6.90·24-s + 7.99·25-s + 15.2·26-s + 1.60·27-s − 3.30·28-s + 4.69·29-s − 19.1·30-s + ⋯ |
| L(s) = 1 | + 1.62·2-s − 1.32·3-s + 1.65·4-s + 1.61·5-s − 2.16·6-s − 0.377·7-s + 1.06·8-s + 0.767·9-s + 2.62·10-s − 2.19·12-s + 1.83·13-s − 0.615·14-s − 2.14·15-s + 0.0756·16-s + 0.654·17-s + 1.24·18-s − 0.688·19-s + 2.66·20-s + 0.502·21-s − 0.562·23-s − 1.41·24-s + 1.59·25-s + 2.98·26-s + 0.308·27-s − 0.624·28-s + 0.872·29-s − 3.49·30-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
| \(L(1)\) |
\(\approx\) |
\(3.283299999\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.283299999\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.30T + 2T^{2} \) |
| 3 | \( 1 + 2.30T + 3T^{2} \) |
| 5 | \( 1 - 3.60T + 5T^{2} \) |
| 13 | \( 1 - 6.60T + 13T^{2} \) |
| 17 | \( 1 - 2.69T + 17T^{2} \) |
| 19 | \( 1 + 3T + 19T^{2} \) |
| 23 | \( 1 + 2.69T + 23T^{2} \) |
| 29 | \( 1 - 4.69T + 29T^{2} \) |
| 31 | \( 1 - T + 31T^{2} \) |
| 37 | \( 1 + 5.21T + 37T^{2} \) |
| 41 | \( 1 - 7T + 41T^{2} \) |
| 43 | \( 1 - 1.69T + 43T^{2} \) |
| 47 | \( 1 + 1.90T + 47T^{2} \) |
| 53 | \( 1 + 12.9T + 53T^{2} \) |
| 59 | \( 1 - 6.69T + 59T^{2} \) |
| 61 | \( 1 + 4.30T + 61T^{2} \) |
| 67 | \( 1 - 8.51T + 67T^{2} \) |
| 71 | \( 1 + 4.30T + 71T^{2} \) |
| 73 | \( 1 + 5T + 73T^{2} \) |
| 79 | \( 1 + 8.30T + 79T^{2} \) |
| 83 | \( 1 - 3T + 83T^{2} \) |
| 89 | \( 1 + 14.7T + 89T^{2} \) |
| 97 | \( 1 + 3.60T + 97T^{2} \) |
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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
−10.61073864217765254209257343317, −9.688662009380172744020290037561, −8.536691510752409555694689771636, −6.77542501109344564354514088839, −6.16501556595527572210903444050, −5.87640632389701092989901285336, −5.13356203063426672119656744949, −4.07393722344825346535239951333, −2.86132881855836292769045034247, −1.47045767534238906276780867183,
1.47045767534238906276780867183, 2.86132881855836292769045034247, 4.07393722344825346535239951333, 5.13356203063426672119656744949, 5.87640632389701092989901285336, 6.16501556595527572210903444050, 6.77542501109344564354514088839, 8.536691510752409555694689771636, 9.688662009380172744020290037561, 10.61073864217765254209257343317
