| L(s) = 1 | + 2.31·2-s + 3.36·4-s − 1.76·5-s + 7-s + 3.15·8-s − 4.09·10-s + 5.91·11-s − 6.94·13-s + 2.31·14-s + 0.580·16-s − 4.59·17-s + 2.11·19-s − 5.94·20-s + 13.6·22-s + 7.36·23-s − 1.87·25-s − 16.0·26-s + 3.36·28-s − 1.96·29-s + 8.41·31-s − 4.96·32-s − 10.6·34-s − 1.76·35-s + 10.4·37-s + 4.89·38-s − 5.57·40-s + 8.05·41-s + ⋯ |
| L(s) = 1 | + 1.63·2-s + 1.68·4-s − 0.790·5-s + 0.377·7-s + 1.11·8-s − 1.29·10-s + 1.78·11-s − 1.92·13-s + 0.618·14-s + 0.145·16-s − 1.11·17-s + 0.484·19-s − 1.32·20-s + 2.91·22-s + 1.53·23-s − 0.374·25-s − 3.15·26-s + 0.635·28-s − 0.365·29-s + 1.51·31-s − 0.877·32-s − 1.82·34-s − 0.298·35-s + 1.71·37-s + 0.793·38-s − 0.881·40-s + 1.25·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.109289538\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.109289538\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 127 | \( 1 + T \) |
| good | 2 | \( 1 - 2.31T + 2T^{2} \) |
| 5 | \( 1 + 1.76T + 5T^{2} \) |
| 11 | \( 1 - 5.91T + 11T^{2} \) |
| 13 | \( 1 + 6.94T + 13T^{2} \) |
| 17 | \( 1 + 4.59T + 17T^{2} \) |
| 19 | \( 1 - 2.11T + 19T^{2} \) |
| 23 | \( 1 - 7.36T + 23T^{2} \) |
| 29 | \( 1 + 1.96T + 29T^{2} \) |
| 31 | \( 1 - 8.41T + 31T^{2} \) |
| 37 | \( 1 - 10.4T + 37T^{2} \) |
| 41 | \( 1 - 8.05T + 41T^{2} \) |
| 43 | \( 1 - 2.89T + 43T^{2} \) |
| 47 | \( 1 - 6.04T + 47T^{2} \) |
| 53 | \( 1 - 0.912T + 53T^{2} \) |
| 59 | \( 1 - 5.80T + 59T^{2} \) |
| 61 | \( 1 - 0.182T + 61T^{2} \) |
| 67 | \( 1 - 6.40T + 67T^{2} \) |
| 71 | \( 1 - 2.69T + 71T^{2} \) |
| 73 | \( 1 + 3.40T + 73T^{2} \) |
| 79 | \( 1 - 1.85T + 79T^{2} \) |
| 83 | \( 1 + 12.7T + 83T^{2} \) |
| 89 | \( 1 - 6.46T + 89T^{2} \) |
| 97 | \( 1 + 11.2T + 97T^{2} \) |
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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
−7.33928314119522005480077642024, −7.12231159012789373125820465604, −6.36338846749758538435735058996, −5.61934829594551713543865299832, −4.60537050867975064121587518882, −4.50041925176373763515509685758, −3.80132171289796665255189618271, −2.82722752007271329331501492567, −2.24641726988333177882300520614, −0.875055115378978367466395587210,
0.875055115378978367466395587210, 2.24641726988333177882300520614, 2.82722752007271329331501492567, 3.80132171289796665255189618271, 4.50041925176373763515509685758, 4.60537050867975064121587518882, 5.61934829594551713543865299832, 6.36338846749758538435735058996, 7.12231159012789373125820465604, 7.33928314119522005480077642024
