L(s) = 1 | + 13.5·2-s + 15.5·3-s + 119.·4-s + 141. i·5-s + 211.·6-s + 238. i·7-s + 749.·8-s + 243·9-s + 1.91e3i·10-s − 253. i·11-s + 1.86e3·12-s + 572.·13-s + 3.23e3i·14-s + 2.19e3i·15-s + 2.50e3·16-s − 197. i·17-s + ⋯ |
L(s) = 1 | + 1.69·2-s + 0.577·3-s + 1.86·4-s + 1.12i·5-s + 0.977·6-s + 0.696i·7-s + 1.46·8-s + 0.333·9-s + 1.91i·10-s − 0.190i·11-s + 1.07·12-s + 0.260·13-s + 1.17i·14-s + 0.651i·15-s + 0.612·16-s − 0.0402i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.777 - 0.628i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.777 - 0.628i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{7}{2})\) |
\(\approx\) |
\(5.37405 + 1.89962i\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.37405 + 1.89962i\) |
\(L(4)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - 15.5T \) |
| 23 | \( 1 + (-9.46e3 + 7.64e3i)T \) |
good | 2 | \( 1 - 13.5T + 64T^{2} \) |
| 5 | \( 1 - 141. iT - 1.56e4T^{2} \) |
| 7 | \( 1 - 238. iT - 1.17e5T^{2} \) |
| 11 | \( 1 + 253. iT - 1.77e6T^{2} \) |
| 13 | \( 1 - 572.T + 4.82e6T^{2} \) |
| 17 | \( 1 + 197. iT - 2.41e7T^{2} \) |
| 19 | \( 1 + 1.21e4iT - 4.70e7T^{2} \) |
| 29 | \( 1 + 6.50e3T + 5.94e8T^{2} \) |
| 31 | \( 1 - 4.46e3T + 8.87e8T^{2} \) |
| 37 | \( 1 + 4.48e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 + 5.65e4T + 4.75e9T^{2} \) |
| 43 | \( 1 - 1.19e5iT - 6.32e9T^{2} \) |
| 47 | \( 1 + 6.06e4T + 1.07e10T^{2} \) |
| 53 | \( 1 + 1.00e4iT - 2.21e10T^{2} \) |
| 59 | \( 1 - 4.84e4T + 4.21e10T^{2} \) |
| 61 | \( 1 - 1.69e5iT - 5.15e10T^{2} \) |
| 67 | \( 1 + 5.51e5iT - 9.04e10T^{2} \) |
| 71 | \( 1 + 2.56e5T + 1.28e11T^{2} \) |
| 73 | \( 1 + 7.23e5T + 1.51e11T^{2} \) |
| 79 | \( 1 + 4.30e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 + 9.00e4iT - 3.26e11T^{2} \) |
| 89 | \( 1 - 8.68e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 - 1.17e6iT - 8.32e11T^{2} \) |
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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
−13.67955752254842433656284723259, −12.88839710144175231456671498579, −11.60602874391799285233013562305, −10.75533761987536002464265341078, −8.985367343542106695061175869400, −7.18771240367531644712035251271, −6.24459617557415929701743047472, −4.78519440965354779046868281883, −3.23319709561231308962916785480, −2.46949990820841808033922840827,
1.54043034270718391783293936165, 3.45351097539944370831891594219, 4.46337739101797154455678940786, 5.64074488232798225928084881228, 7.19073939133684497737689015507, 8.583388078798006461954875952901, 10.16679587403933645072137932286, 11.70997080877629216734820440470, 12.69051308006555939248213891541, 13.38380636133298489676269376850