L(s) = 1 | − 1.72·2-s + 0.971·4-s − 1.06·5-s + 4.32·7-s + 1.77·8-s + 1.83·10-s − 11-s + 6.00·13-s − 7.45·14-s − 4.99·16-s + 2.14·17-s + 2.37·19-s − 1.03·20-s + 1.72·22-s − 1.08·23-s − 3.86·25-s − 10.3·26-s + 4.20·28-s − 7.27·29-s − 6.82·31-s + 5.06·32-s − 3.70·34-s − 4.61·35-s + 7.40·37-s − 4.09·38-s − 1.89·40-s + 4.40·41-s + ⋯ |
L(s) = 1 | − 1.21·2-s + 0.485·4-s − 0.477·5-s + 1.63·7-s + 0.627·8-s + 0.581·10-s − 0.301·11-s + 1.66·13-s − 1.99·14-s − 1.24·16-s + 0.521·17-s + 0.544·19-s − 0.231·20-s + 0.367·22-s − 0.227·23-s − 0.772·25-s − 2.02·26-s + 0.794·28-s − 1.35·29-s − 1.22·31-s + 0.896·32-s − 0.635·34-s − 0.780·35-s + 1.21·37-s − 0.664·38-s − 0.299·40-s + 0.688·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6633 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6633 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.261531869\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.261531869\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| 67 | \( 1 + T \) |
good | 2 | \( 1 + 1.72T + 2T^{2} \) |
| 5 | \( 1 + 1.06T + 5T^{2} \) |
| 7 | \( 1 - 4.32T + 7T^{2} \) |
| 13 | \( 1 - 6.00T + 13T^{2} \) |
| 17 | \( 1 - 2.14T + 17T^{2} \) |
| 19 | \( 1 - 2.37T + 19T^{2} \) |
| 23 | \( 1 + 1.08T + 23T^{2} \) |
| 29 | \( 1 + 7.27T + 29T^{2} \) |
| 31 | \( 1 + 6.82T + 31T^{2} \) |
| 37 | \( 1 - 7.40T + 37T^{2} \) |
| 41 | \( 1 - 4.40T + 41T^{2} \) |
| 43 | \( 1 - 4.27T + 43T^{2} \) |
| 47 | \( 1 - 6.39T + 47T^{2} \) |
| 53 | \( 1 - 11.4T + 53T^{2} \) |
| 59 | \( 1 - 10.1T + 59T^{2} \) |
| 61 | \( 1 - 2.53T + 61T^{2} \) |
| 71 | \( 1 - 12.8T + 71T^{2} \) |
| 73 | \( 1 + 1.68T + 73T^{2} \) |
| 79 | \( 1 - 1.90T + 79T^{2} \) |
| 83 | \( 1 + 6.54T + 83T^{2} \) |
| 89 | \( 1 - 4.54T + 89T^{2} \) |
| 97 | \( 1 + 13.7T + 97T^{2} \) |
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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
−7.983371510901298237510732604646, −7.69334662308351299057597813189, −7.02304150008982281728985576309, −5.71897574066054303894385420191, −5.36505392035207687353125693994, −4.11335919098626236377939213211, −3.88728229630995994476967709601, −2.32985062274896901849616071176, −1.49359039021857240077819772573, −0.77818006451242616331468311451,
0.77818006451242616331468311451, 1.49359039021857240077819772573, 2.32985062274896901849616071176, 3.88728229630995994476967709601, 4.11335919098626236377939213211, 5.36505392035207687353125693994, 5.71897574066054303894385420191, 7.02304150008982281728985576309, 7.69334662308351299057597813189, 7.983371510901298237510732604646