L(s) = 1 | − 0.946·2-s − 1.10·4-s + 2.90·5-s + 7-s + 2.93·8-s − 2.74·10-s + 4.94·11-s + 2.17·13-s − 0.946·14-s − 0.572·16-s + 3.30·17-s − 3.52·19-s − 3.20·20-s − 4.68·22-s − 0.742·23-s + 3.44·25-s − 2.05·26-s − 1.10·28-s − 0.319·29-s − 7.09·31-s − 5.33·32-s − 3.12·34-s + 2.90·35-s + 5.44·37-s + 3.34·38-s + 8.53·40-s − 4.63·41-s + ⋯ |
L(s) = 1 | − 0.669·2-s − 0.552·4-s + 1.29·5-s + 0.377·7-s + 1.03·8-s − 0.869·10-s + 1.49·11-s + 0.603·13-s − 0.252·14-s − 0.143·16-s + 0.800·17-s − 0.809·19-s − 0.717·20-s − 0.998·22-s − 0.154·23-s + 0.688·25-s − 0.403·26-s − 0.208·28-s − 0.0592·29-s − 1.27·31-s − 0.942·32-s − 0.535·34-s + 0.491·35-s + 0.895·37-s + 0.541·38-s + 1.34·40-s − 0.723·41-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
\(L(1)\) |
\(\approx\) |
\(2.079504329\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.079504329\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.946T + 2T^{2} \) |
| 5 | \( 1 - 2.90T + 5T^{2} \) |
| 11 | \( 1 - 4.94T + 11T^{2} \) |
| 13 | \( 1 - 2.17T + 13T^{2} \) |
| 17 | \( 1 - 3.30T + 17T^{2} \) |
| 19 | \( 1 + 3.52T + 19T^{2} \) |
| 23 | \( 1 + 0.742T + 23T^{2} \) |
| 29 | \( 1 + 0.319T + 29T^{2} \) |
| 31 | \( 1 + 7.09T + 31T^{2} \) |
| 37 | \( 1 - 5.44T + 37T^{2} \) |
| 41 | \( 1 + 4.63T + 41T^{2} \) |
| 43 | \( 1 - 2.59T + 43T^{2} \) |
| 47 | \( 1 - 2.05T + 47T^{2} \) |
| 53 | \( 1 - 8.76T + 53T^{2} \) |
| 59 | \( 1 - 12.4T + 59T^{2} \) |
| 61 | \( 1 - 10.4T + 61T^{2} \) |
| 67 | \( 1 - 3.00T + 67T^{2} \) |
| 71 | \( 1 - 15.6T + 71T^{2} \) |
| 73 | \( 1 - 6.60T + 73T^{2} \) |
| 79 | \( 1 - 2.76T + 79T^{2} \) |
| 83 | \( 1 + 11.9T + 83T^{2} \) |
| 89 | \( 1 + 10.9T + 89T^{2} \) |
| 97 | \( 1 + 15.7T + 97T^{2} \) |
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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
−8.102521746377315572677355293983, −7.10069277218890230135116441720, −6.53830972181897371787953375577, −5.66206319067471416109758501887, −5.27963796652946154208203707016, −4.10855367386774790414105994877, −3.76638932305876288798700660115, −2.30549484801610836968924695963, −1.56708833045715430323760860606, −0.885821341012196396600156838799,
0.885821341012196396600156838799, 1.56708833045715430323760860606, 2.30549484801610836968924695963, 3.76638932305876288798700660115, 4.10855367386774790414105994877, 5.27963796652946154208203707016, 5.66206319067471416109758501887, 6.53830972181897371787953375577, 7.10069277218890230135116441720, 8.102521746377315572677355293983