L(s) = 1 | − 4·2-s + 0.233·3-s + 16·4-s − 25·5-s − 0.935·6-s + 203.·7-s − 64·8-s − 242.·9-s + 100·10-s − 151.·11-s + 3.74·12-s + 1.15e3·13-s − 814.·14-s − 5.84·15-s + 256·16-s − 692.·17-s + 971.·18-s − 360.·19-s − 400·20-s + 47.6·21-s + 604.·22-s + 529·23-s − 14.9·24-s + 625·25-s − 4.61e3·26-s − 113.·27-s + 3.25e3·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.0150·3-s + 0.5·4-s − 0.447·5-s − 0.0106·6-s + 1.56·7-s − 0.353·8-s − 0.999·9-s + 0.316·10-s − 0.376·11-s + 0.00750·12-s + 1.89·13-s − 1.11·14-s − 0.00671·15-s + 0.250·16-s − 0.581·17-s + 0.706·18-s − 0.229·19-s − 0.223·20-s + 0.0235·21-s + 0.266·22-s + 0.208·23-s − 0.00530·24-s + 0.200·25-s − 1.33·26-s − 0.0300·27-s + 0.784·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(1.484395079\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.484395079\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 4T \) |
| 5 | \( 1 + 25T \) |
| 23 | \( 1 - 529T \) |
good | 3 | \( 1 - 0.233T + 243T^{2} \) |
| 7 | \( 1 - 203.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 151.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 1.15e3T + 3.71e5T^{2} \) |
| 17 | \( 1 + 692.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 360.T + 2.47e6T^{2} \) |
| 29 | \( 1 + 3.68e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 868.T + 2.86e7T^{2} \) |
| 37 | \( 1 - 4.02e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 6.44e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 6.56e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.28e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.15e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.86e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.67e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.94e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 3.18e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.91e3T + 2.07e9T^{2} \) |
| 79 | \( 1 - 1.00e5T + 3.07e9T^{2} \) |
| 83 | \( 1 - 7.71e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.06e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 3.58e4T + 8.58e9T^{2} \) |
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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
−11.09388844430546426587136207170, −10.79099204512765737190926797396, −9.026577096752565762445732440820, −8.382831298562673535696631739702, −7.75658069396741811043955260953, −6.31234196687001658927626908922, −5.17362544898285878903348890785, −3.73706323688091016719729290349, −2.14777927476108878715770390398, −0.827117208378010247823547064852,
0.827117208378010247823547064852, 2.14777927476108878715770390398, 3.73706323688091016719729290349, 5.17362544898285878903348890785, 6.31234196687001658927626908922, 7.75658069396741811043955260953, 8.382831298562673535696631739702, 9.026577096752565762445732440820, 10.79099204512765737190926797396, 11.09388844430546426587136207170