L(s) = 1 | + 1.78e5i·2-s − 8.55e7i·3-s − 2.34e10·4-s + 1.52e13·6-s − 8.09e13i·7-s − 2.64e15i·8-s − 1.75e15·9-s + 1.95e17·11-s + 2.00e18i·12-s + 2.59e18i·13-s + 1.44e19·14-s + 2.72e20·16-s − 1.52e20i·17-s − 3.13e20i·18-s − 1.39e21·19-s + ⋯ |
L(s) = 1 | + 1.92i·2-s − 1.14i·3-s − 2.72·4-s + 2.21·6-s − 0.920i·7-s − 3.32i·8-s − 0.315·9-s + 1.28·11-s + 3.12i·12-s + 1.08i·13-s + 1.77·14-s + 3.69·16-s − 0.758i·17-s − 0.608i·18-s − 1.10·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(34-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+33/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(17)\) |
\(\approx\) |
\(1.820403983\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.820403983\) |
\(L(\frac{35}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
good | 2 | \( 1 - 1.78e5iT - 8.58e9T^{2} \) |
| 3 | \( 1 + 8.55e7iT - 5.55e15T^{2} \) |
| 7 | \( 1 + 8.09e13iT - 7.73e27T^{2} \) |
| 11 | \( 1 - 1.95e17T + 2.32e34T^{2} \) |
| 13 | \( 1 - 2.59e18iT - 5.75e36T^{2} \) |
| 17 | \( 1 + 1.52e20iT - 4.02e40T^{2} \) |
| 19 | \( 1 + 1.39e21T + 1.58e42T^{2} \) |
| 23 | \( 1 - 3.40e22iT - 8.65e44T^{2} \) |
| 29 | \( 1 - 1.49e24T + 1.81e48T^{2} \) |
| 31 | \( 1 + 1.38e24T + 1.64e49T^{2} \) |
| 37 | \( 1 - 1.15e26iT - 5.63e51T^{2} \) |
| 41 | \( 1 + 6.35e25T + 1.66e53T^{2} \) |
| 43 | \( 1 - 2.64e26iT - 8.02e53T^{2} \) |
| 47 | \( 1 - 1.03e27iT - 1.51e55T^{2} \) |
| 53 | \( 1 + 5.20e28iT - 7.96e56T^{2} \) |
| 59 | \( 1 - 6.17e28T + 2.74e58T^{2} \) |
| 61 | \( 1 + 1.21e29T + 8.23e58T^{2} \) |
| 67 | \( 1 + 2.87e29iT - 1.82e60T^{2} \) |
| 71 | \( 1 + 2.84e30T + 1.23e61T^{2} \) |
| 73 | \( 1 - 2.67e30iT - 3.08e61T^{2} \) |
| 79 | \( 1 - 7.84e30T + 4.18e62T^{2} \) |
| 83 | \( 1 + 1.28e31iT - 2.13e63T^{2} \) |
| 89 | \( 1 - 1.56e32T + 2.13e64T^{2} \) |
| 97 | \( 1 - 4.84e32iT - 3.65e65T^{2} \) |
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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
−11.88407585717827659476738825907, −9.763286711459023318367015771665, −8.672965953819221211431385927046, −7.57822188280177280552188596707, −6.75018016504118560731951358259, −6.39697606974394641979843596040, −4.73856239416812249182759773056, −3.84416119241801270133099970908, −1.49258555100176961134242711108, −0.62082918756665698136625786067,
0.61292131097700950164597956546, 1.84204087753400759778753592811, 2.87585005539811733181318198526, 3.89380041802622521165191720247, 4.56629432792764033702605081449, 5.81750708865092065861956507503, 8.574004083715004413833926071056, 9.126100293122948194051962237043, 10.28686112705090042727918921967, 10.82756889000433769920772541180