L(s) = 1 | − 2.82i·2-s − 21.0·5-s + (−11 − 14.8i)7-s − 22.6i·8-s + 59.5i·10-s − 15.5i·11-s + 29.7i·13-s + (−42.1 + 31.1i)14-s − 64.0·16-s + 63.2·17-s − 89.3i·19-s − 44·22-s + 77.7i·23-s + 318.·25-s + 84.2·26-s + ⋯ |
L(s) = 1 | − 0.999i·2-s − 1.88·5-s + (−0.593 − 0.804i)7-s − 0.999i·8-s + 1.88i·10-s − 0.426i·11-s + 0.635i·13-s + (−0.804 + 0.593i)14-s − 1.00·16-s + 0.901·17-s − 1.07i·19-s − 0.426·22-s + 0.705i·23-s + 2.55·25-s + 0.635·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0204i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0204i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.00804613 - 0.785999i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.00804613 - 0.785999i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (11 + 14.8i)T \) |
good | 2 | \( 1 + 2.82iT - 8T^{2} \) |
| 5 | \( 1 + 21.0T + 125T^{2} \) |
| 11 | \( 1 + 15.5iT - 1.33e3T^{2} \) |
| 13 | \( 1 - 29.7iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 63.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 89.3iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 77.7iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 125. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 238. iT - 2.97e4T^{2} \) |
| 37 | \( 1 + 184T + 5.06e4T^{2} \) |
| 41 | \( 1 - 105.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 190T + 7.95e4T^{2} \) |
| 47 | \( 1 + 42.1T + 1.03e5T^{2} \) |
| 53 | \( 1 - 357. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 84.2T + 2.05e5T^{2} \) |
| 61 | \( 1 + 655. iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 296T + 3.00e5T^{2} \) |
| 71 | \( 1 + 329. iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 804. iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 836T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.22e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 695.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 566. iT - 9.12e5T^{2} \) |
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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
−13.60807334770364290077829177500, −12.42856316904692678483301745266, −11.54574217363783809685834175951, −10.90343172657409847469808742847, −9.522816736914264142086769371306, −7.83296546528896013648912018195, −6.84292784910064500924589121149, −4.18225306365907447239200869850, −3.26560050153144205464053973087, −0.53856234751400809301860077153,
3.32880601330551707161350680948, 5.19318646703473447541040739388, 6.74280033067996113785743085337, 7.81414153030141916024143296395, 8.615087947156589652757809777505, 10.55939556092226575416043890539, 11.96295467126381038910488315068, 12.45640310706611736288809972565, 14.56867982112399747676689084239, 15.14484154402585056385357767080