L(s) = 1 | + 1.56e5i·2-s + 6.96e7i·3-s − 1.57e10·4-s + (−2.14e11 + 2.65e11i)5-s − 1.08e13·6-s − 1.55e14i·7-s − 1.12e15i·8-s + 7.11e14·9-s + (−4.14e16 − 3.34e16i)10-s + 4.13e16·11-s − 1.09e18i·12-s + 2.38e18i·13-s + 2.42e19·14-s + (−1.84e19 − 1.49e19i)15-s + 3.93e19·16-s − 2.91e20i·17-s + ⋯ |
L(s) = 1 | + 1.68i·2-s + 0.933i·3-s − 1.83·4-s + (−0.627 + 0.778i)5-s − 1.57·6-s − 1.76i·7-s − 1.40i·8-s + 0.128·9-s + (−1.31 − 1.05i)10-s + 0.271·11-s − 1.71i·12-s + 0.993i·13-s + 2.97·14-s + (−0.726 − 0.586i)15-s + 0.533·16-s − 1.45i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.627 - 0.778i)\, \overline{\Lambda}(34-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+33/2) \, L(s)\cr =\mathstrut & (0.627 - 0.778i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(17)\) |
\(\approx\) |
\(0.8440923622\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8440923622\) |
\(L(\frac{35}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (2.14e11 - 2.65e11i)T \) |
good | 2 | \( 1 - 1.56e5iT - 8.58e9T^{2} \) |
| 3 | \( 1 - 6.96e7iT - 5.55e15T^{2} \) |
| 7 | \( 1 + 1.55e14iT - 7.73e27T^{2} \) |
| 11 | \( 1 - 4.13e16T + 2.32e34T^{2} \) |
| 13 | \( 1 - 2.38e18iT - 5.75e36T^{2} \) |
| 17 | \( 1 + 2.91e20iT - 4.02e40T^{2} \) |
| 19 | \( 1 - 3.61e20T + 1.58e42T^{2} \) |
| 23 | \( 1 + 3.81e22iT - 8.65e44T^{2} \) |
| 29 | \( 1 + 1.65e24T + 1.81e48T^{2} \) |
| 31 | \( 1 + 2.76e24T + 1.64e49T^{2} \) |
| 37 | \( 1 + 8.70e24iT - 5.63e51T^{2} \) |
| 41 | \( 1 - 2.69e26T + 1.66e53T^{2} \) |
| 43 | \( 1 - 1.33e26iT - 8.02e53T^{2} \) |
| 47 | \( 1 + 2.88e27iT - 1.51e55T^{2} \) |
| 53 | \( 1 - 2.92e28iT - 7.96e56T^{2} \) |
| 59 | \( 1 + 1.51e29T + 2.74e58T^{2} \) |
| 61 | \( 1 - 2.88e29T + 8.23e58T^{2} \) |
| 67 | \( 1 + 8.72e29iT - 1.82e60T^{2} \) |
| 71 | \( 1 - 3.08e30T + 1.23e61T^{2} \) |
| 73 | \( 1 + 2.41e30iT - 3.08e61T^{2} \) |
| 79 | \( 1 - 3.21e31T + 4.18e62T^{2} \) |
| 83 | \( 1 + 7.79e31iT - 2.13e63T^{2} \) |
| 89 | \( 1 + 1.41e32T + 2.13e64T^{2} \) |
| 97 | \( 1 - 7.13e32iT - 3.65e65T^{2} \) |
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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
−16.12979880240380770564325193103, −14.72145314275537491238468675049, −13.84390223253056789807467839791, −10.93453396206855901609794914814, −9.450136055534139918412846644117, −7.48156452421808211893012611057, −6.80807642323918498788538557641, −4.64075123348243738719319160034, −3.84137935240166514637458185348, −0.28932661428998160676810426312,
1.18487300014509095919805226043, 2.09644892196567862424782551314, 3.63497276296200495581833618792, 5.50374370778754642935691524640, 8.121260604441988462987898006669, 9.382932015746195596324373729808, 11.40643986180148347604205165396, 12.49515995756446240034957728560, 12.92383368359655645332987764219, 15.37280275031650888822394203570