L(s) = 1 | − 2.78·2-s − 26.7·3-s − 24.2·4-s − 25·5-s + 74.6·6-s − 117.·7-s + 156.·8-s + 474.·9-s + 69.6·10-s − 121·11-s + 649.·12-s − 1.03e3·13-s + 328.·14-s + 669.·15-s + 338.·16-s + 1.51e3·17-s − 1.32e3·18-s + 361·19-s + 605.·20-s + 3.15e3·21-s + 337.·22-s + 1.69e3·23-s − 4.19e3·24-s + 625·25-s + 2.88e3·26-s − 6.19e3·27-s + 2.85e3·28-s + ⋯ |
L(s) = 1 | − 0.492·2-s − 1.71·3-s − 0.757·4-s − 0.447·5-s + 0.846·6-s − 0.909·7-s + 0.865·8-s + 1.95·9-s + 0.220·10-s − 0.301·11-s + 1.30·12-s − 1.69·13-s + 0.448·14-s + 0.768·15-s + 0.330·16-s + 1.27·17-s − 0.961·18-s + 0.229·19-s + 0.338·20-s + 1.56·21-s + 0.148·22-s + 0.667·23-s − 1.48·24-s + 0.200·25-s + 0.836·26-s − 1.63·27-s + 0.689·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.3371538951\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3371538951\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + 25T \) |
| 11 | \( 1 + 121T \) |
| 19 | \( 1 - 361T \) |
good | 2 | \( 1 + 2.78T + 32T^{2} \) |
| 3 | \( 1 + 26.7T + 243T^{2} \) |
| 7 | \( 1 + 117.T + 1.68e4T^{2} \) |
| 13 | \( 1 + 1.03e3T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.51e3T + 1.41e6T^{2} \) |
| 23 | \( 1 - 1.69e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 4.98e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 6.81e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 840.T + 6.93e7T^{2} \) |
| 41 | \( 1 + 4.17e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 6.31e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.47e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.41e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 4.14e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.98e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 6.13e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 6.01e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.42e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 3.26e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.93e3T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.44e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.98e4T + 8.58e9T^{2} \) |
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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
−9.613832720862587536655758065324, −8.300242219905904869414763619504, −7.36752038700786897524586130613, −6.76602781915938949456628218229, −5.61467486788051575201457821394, −5.00461835249379915169013816868, −4.26453259039238897813000245729, −2.93749784307262355008325067685, −1.06070342138379016538987545561, −0.38235683096127059204238013286,
0.38235683096127059204238013286, 1.06070342138379016538987545561, 2.93749784307262355008325067685, 4.26453259039238897813000245729, 5.00461835249379915169013816868, 5.61467486788051575201457821394, 6.76602781915938949456628218229, 7.36752038700786897524586130613, 8.300242219905904869414763619504, 9.613832720862587536655758065324