| L(s) = 1 | − 2.63·2-s + 0.519·3-s + 4.96·4-s − 5-s − 1.37·6-s − 7.83·8-s − 2.73·9-s + 2.63·10-s + 3.94·11-s + 2.57·12-s + 6.01·13-s − 0.519·15-s + 10.7·16-s − 0.629·17-s + 7.20·18-s − 3.88·19-s − 4.96·20-s − 10.4·22-s + 7.60·23-s − 4.06·24-s + 25-s − 15.8·26-s − 2.97·27-s + 29-s + 1.37·30-s + 6.11·31-s − 12.7·32-s + ⋯ |
| L(s) = 1 | − 1.86·2-s + 0.299·3-s + 2.48·4-s − 0.447·5-s − 0.559·6-s − 2.77·8-s − 0.910·9-s + 0.834·10-s + 1.19·11-s + 0.744·12-s + 1.66·13-s − 0.134·15-s + 2.68·16-s − 0.152·17-s + 1.69·18-s − 0.891·19-s − 1.11·20-s − 2.22·22-s + 1.58·23-s − 0.830·24-s + 0.200·25-s − 3.11·26-s − 0.572·27-s + 0.185·29-s + 0.250·30-s + 1.09·31-s − 2.24·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 + 2.63T + 2T^{2} \) |
| 3 | \( 1 - 0.519T + 3T^{2} \) |
| 11 | \( 1 - 3.94T + 11T^{2} \) |
| 13 | \( 1 - 6.01T + 13T^{2} \) |
| 17 | \( 1 + 0.629T + 17T^{2} \) |
| 19 | \( 1 + 3.88T + 19T^{2} \) |
| 23 | \( 1 - 7.60T + 23T^{2} \) |
| 31 | \( 1 - 6.11T + 31T^{2} \) |
| 37 | \( 1 + 9.02T + 37T^{2} \) |
| 41 | \( 1 + 11.8T + 41T^{2} \) |
| 43 | \( 1 + 1.37T + 43T^{2} \) |
| 47 | \( 1 + 9.83T + 47T^{2} \) |
| 53 | \( 1 + 12.3T + 53T^{2} \) |
| 59 | \( 1 + 4.81T + 59T^{2} \) |
| 61 | \( 1 + 4.60T + 61T^{2} \) |
| 67 | \( 1 - 8.74T + 67T^{2} \) |
| 71 | \( 1 + 11.5T + 71T^{2} \) |
| 73 | \( 1 + 6.03T + 73T^{2} \) |
| 79 | \( 1 - 3.37T + 79T^{2} \) |
| 83 | \( 1 - 5.70T + 83T^{2} \) |
| 89 | \( 1 + 0.657T + 89T^{2} \) |
| 97 | \( 1 + 13.4T + 97T^{2} \) |
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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
−8.030533300949850286274413287794, −6.86822711915315218412637298995, −6.61377277762898745555338334178, −5.95515691282662980164949501361, −4.72070927442840641291721746453, −3.43831921671319697923391668570, −3.11252323291366390725809240384, −1.81553639954435665206336605845, −1.17968304368599453986336931370, 0,
1.17968304368599453986336931370, 1.81553639954435665206336605845, 3.11252323291366390725809240384, 3.43831921671319697923391668570, 4.72070927442840641291721746453, 5.95515691282662980164949501361, 6.61377277762898745555338334178, 6.86822711915315218412637298995, 8.030533300949850286274413287794
