| L(s) = 1 | + 1.68·2-s − 3-s + 0.855·4-s + 2.75·5-s − 1.68·6-s − 7-s − 1.93·8-s + 9-s + 4.65·10-s + 3.55·11-s − 0.855·12-s + 4.67·13-s − 1.68·14-s − 2.75·15-s − 4.97·16-s + 4.02·17-s + 1.68·18-s − 1.08·19-s + 2.35·20-s + 21-s + 6.00·22-s − 7.04·23-s + 1.93·24-s + 2.58·25-s + 7.89·26-s − 27-s − 0.855·28-s + ⋯ |
| L(s) = 1 | + 1.19·2-s − 0.577·3-s + 0.427·4-s + 1.23·5-s − 0.689·6-s − 0.377·7-s − 0.683·8-s + 0.333·9-s + 1.47·10-s + 1.07·11-s − 0.246·12-s + 1.29·13-s − 0.451·14-s − 0.710·15-s − 1.24·16-s + 0.977·17-s + 0.398·18-s − 0.249·19-s + 0.526·20-s + 0.218·21-s + 1.28·22-s − 1.46·23-s + 0.394·24-s + 0.516·25-s + 1.54·26-s − 0.192·27-s − 0.161·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.500804970\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.500804970\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 127 | \( 1 - T \) |
| good | 2 | \( 1 - 1.68T + 2T^{2} \) |
| 5 | \( 1 - 2.75T + 5T^{2} \) |
| 11 | \( 1 - 3.55T + 11T^{2} \) |
| 13 | \( 1 - 4.67T + 13T^{2} \) |
| 17 | \( 1 - 4.02T + 17T^{2} \) |
| 19 | \( 1 + 1.08T + 19T^{2} \) |
| 23 | \( 1 + 7.04T + 23T^{2} \) |
| 29 | \( 1 - 9.79T + 29T^{2} \) |
| 31 | \( 1 + 2.65T + 31T^{2} \) |
| 37 | \( 1 - 10.4T + 37T^{2} \) |
| 41 | \( 1 + 9.96T + 41T^{2} \) |
| 43 | \( 1 + 6.65T + 43T^{2} \) |
| 47 | \( 1 - 6.75T + 47T^{2} \) |
| 53 | \( 1 + 3.41T + 53T^{2} \) |
| 59 | \( 1 - 8.15T + 59T^{2} \) |
| 61 | \( 1 - 15.2T + 61T^{2} \) |
| 67 | \( 1 + 10.7T + 67T^{2} \) |
| 71 | \( 1 - 5.93T + 71T^{2} \) |
| 73 | \( 1 - 8.88T + 73T^{2} \) |
| 79 | \( 1 - 10.1T + 79T^{2} \) |
| 83 | \( 1 - 12.6T + 83T^{2} \) |
| 89 | \( 1 + 6.83T + 89T^{2} \) |
| 97 | \( 1 - 13.2T + 97T^{2} \) |
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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
−8.945217607892365564769305440400, −8.152309536381797472210411562478, −6.62066431451295229142899326632, −6.35442549380062158545304457075, −5.79106631687424617302867919414, −5.08202395378065200425787355676, −4.05271795927969010147581203003, −3.48153763859176648547309181055, −2.25176419192868796334754333185, −1.08513207198477551738255770237,
1.08513207198477551738255770237, 2.25176419192868796334754333185, 3.48153763859176648547309181055, 4.05271795927969010147581203003, 5.08202395378065200425787355676, 5.79106631687424617302867919414, 6.35442549380062158545304457075, 6.62066431451295229142899326632, 8.152309536381797472210411562478, 8.945217607892365564769305440400
