| L(s) = 1 | + (−5.97 + 3.83i)2-s + (−2.21 + 15.4i)3-s + (−5.66 + 12.4i)4-s + (7.37 + 25.1i)5-s + (−45.9 − 100. i)6-s + (−258. + 223. i)7-s + (−78.4 − 545. i)8-s + (−233. − 68.4i)9-s + (−140. − 121. i)10-s + (−783. + 1.21e3i)11-s + (−178. − 114. i)12-s + (−412. + 475. i)13-s + (682. − 2.32e3i)14-s + (−403. + 58.0i)15-s + (1.98e3 + 2.29e3i)16-s + (−1.05e3 + 482. i)17-s + ⋯ |
| L(s) = 1 | + (−0.746 + 0.479i)2-s + (−0.0821 + 0.571i)3-s + (−0.0884 + 0.193i)4-s + (0.0589 + 0.200i)5-s + (−0.212 − 0.465i)6-s + (−0.752 + 0.652i)7-s + (−0.153 − 1.06i)8-s + (−0.319 − 0.0939i)9-s + (−0.140 − 0.121i)10-s + (−0.588 + 0.916i)11-s + (−0.103 − 0.0664i)12-s + (−0.187 + 0.216i)13-s + (0.248 − 0.847i)14-s + (−0.119 + 0.0172i)15-s + (0.485 + 0.560i)16-s + (−0.214 + 0.0981i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.372 + 0.928i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.372 + 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.00781811 - 0.00528776i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00781811 - 0.00528776i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (2.21 - 15.4i)T \) |
| 23 | \( 1 + (-1.19e4 + 2.09e3i)T \) |
| good | 2 | \( 1 + (5.97 - 3.83i)T + (26.5 - 58.2i)T^{2} \) |
| 5 | \( 1 + (-7.37 - 25.1i)T + (-1.31e4 + 8.44e3i)T^{2} \) |
| 7 | \( 1 + (258. - 223. i)T + (1.67e4 - 1.16e5i)T^{2} \) |
| 11 | \( 1 + (783. - 1.21e3i)T + (-7.35e5 - 1.61e6i)T^{2} \) |
| 13 | \( 1 + (412. - 475. i)T + (-6.86e5 - 4.77e6i)T^{2} \) |
| 17 | \( 1 + (1.05e3 - 482. i)T + (1.58e7 - 1.82e7i)T^{2} \) |
| 19 | \( 1 + (3.98e3 + 1.81e3i)T + (3.08e7 + 3.55e7i)T^{2} \) |
| 29 | \( 1 + (-5.91e3 - 1.29e4i)T + (-3.89e8 + 4.49e8i)T^{2} \) |
| 31 | \( 1 + (8.35e3 + 5.81e4i)T + (-8.51e8 + 2.50e8i)T^{2} \) |
| 37 | \( 1 + (69.9 - 238. i)T + (-2.15e9 - 1.38e9i)T^{2} \) |
| 41 | \( 1 + (-1.00e5 + 2.93e4i)T + (3.99e9 - 2.56e9i)T^{2} \) |
| 43 | \( 1 + (8.05e4 + 1.15e4i)T + (6.06e9 + 1.78e9i)T^{2} \) |
| 47 | \( 1 - 9.10e3T + 1.07e10T^{2} \) |
| 53 | \( 1 + (1.55e5 - 1.34e5i)T + (3.15e9 - 2.19e10i)T^{2} \) |
| 59 | \( 1 + (1.75e5 - 2.02e5i)T + (-6.00e9 - 4.17e10i)T^{2} \) |
| 61 | \( 1 + (7.49e3 - 1.07e3i)T + (4.94e10 - 1.45e10i)T^{2} \) |
| 67 | \( 1 + (1.90e5 + 2.96e5i)T + (-3.75e10 + 8.22e10i)T^{2} \) |
| 71 | \( 1 + (1.68e3 - 1.08e3i)T + (5.32e10 - 1.16e11i)T^{2} \) |
| 73 | \( 1 + (5.27e4 - 1.15e5i)T + (-9.91e10 - 1.14e11i)T^{2} \) |
| 79 | \( 1 + (4.03e5 + 3.49e5i)T + (3.45e10 + 2.40e11i)T^{2} \) |
| 83 | \( 1 + (-1.87e5 + 6.38e5i)T + (-2.75e11 - 1.76e11i)T^{2} \) |
| 89 | \( 1 + (-5.24e5 - 7.53e4i)T + (4.76e11 + 1.40e11i)T^{2} \) |
| 97 | \( 1 + (3.66e5 + 1.24e6i)T + (-7.00e11 + 4.50e11i)T^{2} \) |
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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
−14.87990872449849644413400725580, −13.11289241478039471185325700162, −12.33179155821650537998093785970, −10.70209226140913848861609424667, −9.590684738327411327105074623146, −8.869743792805861896630977274918, −7.45535965745764065773035415271, −6.25284201173612326533024884151, −4.49160172806042498893216234515, −2.78834252865443963884487801167,
0.00544145943414850594305468454, 1.16477259142160455770328986360, 2.96966701582278301432843027335, 5.21104915297169174142862573026, 6.63985149579653009698507060060, 8.121818057949091897985819726936, 9.178555827488119075321494300937, 10.41640894413260915164265849008, 11.16418688231930892615478180565, 12.67088453992988574050889558053
