L(s) = 1 | − 2.22·2-s + 0.328·3-s + 2.93·4-s + 2.84·5-s − 0.729·6-s − 1.80·7-s − 2.08·8-s − 2.89·9-s − 6.32·10-s + 1.40·11-s + 0.964·12-s + 5.09·13-s + 4.00·14-s + 0.934·15-s − 1.24·16-s − 0.501·17-s + 6.42·18-s + 7.81·19-s + 8.35·20-s − 0.591·21-s − 3.13·22-s − 23-s − 0.684·24-s + 3.09·25-s − 11.3·26-s − 1.93·27-s − 5.29·28-s + ⋯ |
L(s) = 1 | − 1.57·2-s + 0.189·3-s + 1.46·4-s + 1.27·5-s − 0.297·6-s − 0.680·7-s − 0.736·8-s − 0.964·9-s − 1.99·10-s + 0.424·11-s + 0.278·12-s + 1.41·13-s + 1.06·14-s + 0.241·15-s − 0.311·16-s − 0.121·17-s + 1.51·18-s + 1.79·19-s + 1.86·20-s − 0.129·21-s − 0.667·22-s − 0.208·23-s − 0.139·24-s + 0.618·25-s − 2.22·26-s − 0.372·27-s − 0.999·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8892867945\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8892867945\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
good | 2 | \( 1 + 2.22T + 2T^{2} \) |
| 3 | \( 1 - 0.328T + 3T^{2} \) |
| 5 | \( 1 - 2.84T + 5T^{2} \) |
| 7 | \( 1 + 1.80T + 7T^{2} \) |
| 11 | \( 1 - 1.40T + 11T^{2} \) |
| 13 | \( 1 - 5.09T + 13T^{2} \) |
| 17 | \( 1 + 0.501T + 17T^{2} \) |
| 19 | \( 1 - 7.81T + 19T^{2} \) |
| 31 | \( 1 + 7.79T + 31T^{2} \) |
| 37 | \( 1 - 6.33T + 37T^{2} \) |
| 41 | \( 1 - 4.39T + 41T^{2} \) |
| 43 | \( 1 + 8.99T + 43T^{2} \) |
| 47 | \( 1 - 11.9T + 47T^{2} \) |
| 53 | \( 1 - 8.71T + 53T^{2} \) |
| 59 | \( 1 - 11.5T + 59T^{2} \) |
| 61 | \( 1 + 7.18T + 61T^{2} \) |
| 67 | \( 1 - 9.87T + 67T^{2} \) |
| 71 | \( 1 + 0.384T + 71T^{2} \) |
| 73 | \( 1 - 9.35T + 73T^{2} \) |
| 79 | \( 1 + 4.12T + 79T^{2} \) |
| 83 | \( 1 + 10.6T + 83T^{2} \) |
| 89 | \( 1 - 5.51T + 89T^{2} \) |
| 97 | \( 1 - 9.04T + 97T^{2} \) |
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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
−10.18534726717053082002440896735, −9.436442228477065328438483353510, −9.064208741518015062528660657665, −8.224047645305079268972768486033, −7.14462514057110105501870016354, −6.18932709977296047843303025425, −5.54658854219835834837691981048, −3.51632012976807128916737389880, −2.30768569437207625735976389202, −1.06157068357453934421540025652,
1.06157068357453934421540025652, 2.30768569437207625735976389202, 3.51632012976807128916737389880, 5.54658854219835834837691981048, 6.18932709977296047843303025425, 7.14462514057110105501870016354, 8.224047645305079268972768486033, 9.064208741518015062528660657665, 9.436442228477065328438483353510, 10.18534726717053082002440896735