L(s) = 1 | + 0.702·2-s − 1.50·4-s − 2.41·5-s + 0.380·7-s − 2.46·8-s − 1.69·10-s + 11-s + 2.87·13-s + 0.267·14-s + 1.28·16-s + 1.85·17-s − 2.38·19-s + 3.63·20-s + 0.702·22-s − 5.66·23-s + 0.811·25-s + 2.01·26-s − 0.573·28-s + 2.60·29-s − 4.67·31-s + 5.82·32-s + 1.30·34-s − 0.917·35-s − 10.7·37-s − 1.67·38-s + 5.93·40-s − 0.412·41-s + ⋯ |
L(s) = 1 | + 0.496·2-s − 0.753·4-s − 1.07·5-s + 0.143·7-s − 0.871·8-s − 0.535·10-s + 0.301·11-s + 0.796·13-s + 0.0714·14-s + 0.320·16-s + 0.449·17-s − 0.547·19-s + 0.811·20-s + 0.149·22-s − 1.18·23-s + 0.162·25-s + 0.396·26-s − 0.108·28-s + 0.483·29-s − 0.840·31-s + 1.03·32-s + 0.223·34-s − 0.155·35-s − 1.77·37-s − 0.272·38-s + 0.939·40-s − 0.0644·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.147383594\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.147383594\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 - T \) |
| 61 | \( 1 + T \) |
good | 2 | \( 1 - 0.702T + 2T^{2} \) |
| 5 | \( 1 + 2.41T + 5T^{2} \) |
| 7 | \( 1 - 0.380T + 7T^{2} \) |
| 13 | \( 1 - 2.87T + 13T^{2} \) |
| 17 | \( 1 - 1.85T + 17T^{2} \) |
| 19 | \( 1 + 2.38T + 19T^{2} \) |
| 23 | \( 1 + 5.66T + 23T^{2} \) |
| 29 | \( 1 - 2.60T + 29T^{2} \) |
| 31 | \( 1 + 4.67T + 31T^{2} \) |
| 37 | \( 1 + 10.7T + 37T^{2} \) |
| 41 | \( 1 + 0.412T + 41T^{2} \) |
| 43 | \( 1 + 2.52T + 43T^{2} \) |
| 47 | \( 1 - 7.69T + 47T^{2} \) |
| 53 | \( 1 - 3.19T + 53T^{2} \) |
| 59 | \( 1 - 2.95T + 59T^{2} \) |
| 67 | \( 1 - 8.09T + 67T^{2} \) |
| 71 | \( 1 + 15.0T + 71T^{2} \) |
| 73 | \( 1 - 12.1T + 73T^{2} \) |
| 79 | \( 1 + 11.3T + 79T^{2} \) |
| 83 | \( 1 - 13.0T + 83T^{2} \) |
| 89 | \( 1 - 6.76T + 89T^{2} \) |
| 97 | \( 1 - 6.74T + 97T^{2} \) |
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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
−8.206086535154702371778552268694, −7.45770392840612334214438951471, −6.56930125735301914984570591350, −5.82654815437674548105567404525, −5.11569197480808407984153001787, −4.24602776328073767912737091199, −3.79316875034273049117567609648, −3.22167679657598500593694251708, −1.83121768819141626945028923268, −0.52207521251493647622584291866,
0.52207521251493647622584291866, 1.83121768819141626945028923268, 3.22167679657598500593694251708, 3.79316875034273049117567609648, 4.24602776328073767912737091199, 5.11569197480808407984153001787, 5.82654815437674548105567404525, 6.56930125735301914984570591350, 7.45770392840612334214438951471, 8.206086535154702371778552268694