| L(s) = 1 | + 2.12·2-s − 3.12·3-s + 2.51·4-s + 0.484·5-s − 6.64·6-s − 7-s + 1.09·8-s + 6.76·9-s + 1.03·10-s − 7.85·12-s − 5.60·13-s − 2.12·14-s − 1.51·15-s − 2.70·16-s − 5.60·17-s + 14.3·18-s + 5.28·19-s + 1.21·20-s + 3.12·21-s + 2.48·23-s − 3.42·24-s − 4.76·25-s − 11.9·26-s − 11.7·27-s − 2.51·28-s − 5.28·29-s − 3.21·30-s + ⋯ |
| L(s) = 1 | + 1.50·2-s − 1.80·3-s + 1.25·4-s + 0.216·5-s − 2.71·6-s − 0.377·7-s + 0.387·8-s + 2.25·9-s + 0.325·10-s − 2.26·12-s − 1.55·13-s − 0.567·14-s − 0.391·15-s − 0.676·16-s − 1.36·17-s + 3.38·18-s + 1.21·19-s + 0.272·20-s + 0.681·21-s + 0.518·23-s − 0.698·24-s − 0.952·25-s − 2.33·26-s − 2.26·27-s − 0.475·28-s − 0.980·29-s − 0.587·30-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 2.12T + 2T^{2} \) |
| 3 | \( 1 + 3.12T + 3T^{2} \) |
| 5 | \( 1 - 0.484T + 5T^{2} \) |
| 13 | \( 1 + 5.60T + 13T^{2} \) |
| 17 | \( 1 + 5.60T + 17T^{2} \) |
| 19 | \( 1 - 5.28T + 19T^{2} \) |
| 23 | \( 1 - 2.48T + 23T^{2} \) |
| 29 | \( 1 + 5.28T + 29T^{2} \) |
| 31 | \( 1 + 7.12T + 31T^{2} \) |
| 37 | \( 1 + 0.235T + 37T^{2} \) |
| 41 | \( 1 + 2.39T + 41T^{2} \) |
| 43 | \( 1 + 1.03T + 43T^{2} \) |
| 47 | \( 1 + 1.60T + 47T^{2} \) |
| 53 | \( 1 + 3.03T + 53T^{2} \) |
| 59 | \( 1 - 3.12T + 59T^{2} \) |
| 61 | \( 1 - 2.39T + 61T^{2} \) |
| 67 | \( 1 - 10.0T + 67T^{2} \) |
| 71 | \( 1 - 12.0T + 71T^{2} \) |
| 73 | \( 1 + 2.39T + 73T^{2} \) |
| 79 | \( 1 + 9.03T + 79T^{2} \) |
| 83 | \( 1 - 3.21T + 83T^{2} \) |
| 89 | \( 1 + 1.26T + 89T^{2} \) |
| 97 | \( 1 + 8.79T + 97T^{2} \) |
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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
−9.991649855514483384921005940984, −9.323969163944463819598812375727, −7.32708875773608493986839165494, −6.84785483325004550285318440340, −5.92419631713016693379617853779, −5.26173842224527021115874685524, −4.72407829068804524491932456627, −3.66764501922848674944150779463, −2.14632884657505271210767399285, 0,
2.14632884657505271210767399285, 3.66764501922848674944150779463, 4.72407829068804524491932456627, 5.26173842224527021115874685524, 5.92419631713016693379617853779, 6.84785483325004550285318440340, 7.32708875773608493986839165494, 9.323969163944463819598812375727, 9.991649855514483384921005940984
