L(s) = 1 | + 2.51·2-s − 3-s + 4.33·4-s − 2.41·5-s − 2.51·6-s + 5.87·8-s + 9-s − 6.08·10-s + 1.24·11-s − 4.33·12-s − 2.06·13-s + 2.41·15-s + 6.11·16-s + 3.31·17-s + 2.51·18-s + 3.61·19-s − 10.4·20-s + 3.14·22-s − 23-s − 5.87·24-s + 0.848·25-s − 5.20·26-s − 27-s + 4.27·29-s + 6.08·30-s + 7.97·31-s + 3.64·32-s + ⋯ |
L(s) = 1 | + 1.77·2-s − 0.577·3-s + 2.16·4-s − 1.08·5-s − 1.02·6-s + 2.07·8-s + 0.333·9-s − 1.92·10-s + 0.376·11-s − 1.25·12-s − 0.573·13-s + 0.624·15-s + 1.52·16-s + 0.802·17-s + 0.593·18-s + 0.830·19-s − 2.34·20-s + 0.669·22-s − 0.208·23-s − 1.19·24-s + 0.169·25-s − 1.02·26-s − 0.192·27-s + 0.794·29-s + 1.11·30-s + 1.43·31-s + 0.644·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.097025134\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.097025134\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 23 | \( 1 + T \) |
good | 2 | \( 1 - 2.51T + 2T^{2} \) |
| 5 | \( 1 + 2.41T + 5T^{2} \) |
| 11 | \( 1 - 1.24T + 11T^{2} \) |
| 13 | \( 1 + 2.06T + 13T^{2} \) |
| 17 | \( 1 - 3.31T + 17T^{2} \) |
| 19 | \( 1 - 3.61T + 19T^{2} \) |
| 29 | \( 1 - 4.27T + 29T^{2} \) |
| 31 | \( 1 - 7.97T + 31T^{2} \) |
| 37 | \( 1 - 5.09T + 37T^{2} \) |
| 41 | \( 1 - 0.885T + 41T^{2} \) |
| 43 | \( 1 - 9.94T + 43T^{2} \) |
| 47 | \( 1 - 5.69T + 47T^{2} \) |
| 53 | \( 1 + 2.94T + 53T^{2} \) |
| 59 | \( 1 + 2.02T + 59T^{2} \) |
| 61 | \( 1 - 7.38T + 61T^{2} \) |
| 67 | \( 1 - 8.28T + 67T^{2} \) |
| 71 | \( 1 + 8.27T + 71T^{2} \) |
| 73 | \( 1 - 11.9T + 73T^{2} \) |
| 79 | \( 1 + 15.2T + 79T^{2} \) |
| 83 | \( 1 + 15.7T + 83T^{2} \) |
| 89 | \( 1 - 16.2T + 89T^{2} \) |
| 97 | \( 1 + 16.6T + 97T^{2} \) |
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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
−8.231926437194970665443901001414, −7.51433967596099977937615391456, −6.96817959241869282686052403541, −6.09725066243365167913909433919, −5.49994702310512144849380476982, −4.60555035069687604528927604374, −4.17757838461921932182823408996, −3.32649474693494475558368403713, −2.51254438832606926345658424786, −0.977747314436821742847519639486,
0.977747314436821742847519639486, 2.51254438832606926345658424786, 3.32649474693494475558368403713, 4.17757838461921932182823408996, 4.60555035069687604528927604374, 5.49994702310512144849380476982, 6.09725066243365167913909433919, 6.96817959241869282686052403541, 7.51433967596099977937615391456, 8.231926437194970665443901001414