L(s) = 1 | − 6.12e4i·2-s − 1.16e8i·3-s + 4.83e9·4-s − 7.12e12·6-s + 1.52e14i·7-s − 8.22e14i·8-s − 7.98e15·9-s − 2.70e17·11-s − 5.63e17i·12-s + 3.01e18i·13-s + 9.35e18·14-s − 8.79e18·16-s − 5.13e19i·17-s + 4.89e20i·18-s + 2.04e21·19-s + ⋯ |
L(s) = 1 | − 0.660i·2-s − 1.56i·3-s + 0.563·4-s − 1.03·6-s + 1.73i·7-s − 1.03i·8-s − 1.43·9-s − 1.77·11-s − 0.879i·12-s + 1.25i·13-s + 1.14·14-s − 0.119·16-s − 0.255i·17-s + 0.949i·18-s + 1.62·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(34-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+33/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(17)\) |
\(\approx\) |
\(2.095258299\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.095258299\) |
\(L(\frac{35}{2})\) |
|
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 | \( 1 + 6.12e4iT - 8.58e9T^{2} \) |
| 3 | \( 1 + 1.16e8iT - 5.55e15T^{2} \) |
| 7 | \( 1 - 1.52e14iT - 7.73e27T^{2} \) |
| 11 | \( 1 + 2.70e17T + 2.32e34T^{2} \) |
| 13 | \( 1 - 3.01e18iT - 5.75e36T^{2} \) |
| 17 | \( 1 + 5.13e19iT - 4.02e40T^{2} \) |
| 19 | \( 1 - 2.04e21T + 1.58e42T^{2} \) |
| 23 | \( 1 + 2.04e22iT - 8.65e44T^{2} \) |
| 29 | \( 1 + 1.42e24T + 1.81e48T^{2} \) |
| 31 | \( 1 - 5.31e24T + 1.64e49T^{2} \) |
| 37 | \( 1 - 5.96e25iT - 5.63e51T^{2} \) |
| 41 | \( 1 - 2.10e26T + 1.66e53T^{2} \) |
| 43 | \( 1 + 3.53e26iT - 8.02e53T^{2} \) |
| 47 | \( 1 + 1.08e27iT - 1.51e55T^{2} \) |
| 53 | \( 1 + 1.51e28iT - 7.96e56T^{2} \) |
| 59 | \( 1 - 3.91e28T + 2.74e58T^{2} \) |
| 61 | \( 1 - 1.55e29T + 8.23e58T^{2} \) |
| 67 | \( 1 + 5.34e29iT - 1.82e60T^{2} \) |
| 71 | \( 1 - 4.86e30T + 1.23e61T^{2} \) |
| 73 | \( 1 + 5.33e30iT - 3.08e61T^{2} \) |
| 79 | \( 1 + 2.81e30T + 4.18e62T^{2} \) |
| 83 | \( 1 - 2.25e31iT - 2.13e63T^{2} \) |
| 89 | \( 1 + 5.60e31T + 2.13e64T^{2} \) |
| 97 | \( 1 + 2.05e32iT - 3.65e65T^{2} \) |
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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
−11.34721090000043876322774895557, −9.683770855087540746338928973752, −8.302540934229381030014736755318, −7.29443595540040013522289699418, −6.26226274250381080208421721363, −5.24877027750353284022287661059, −2.93253269696632477143725226089, −2.37880996375935058269234894904, −1.67647015058534950758777216735, −0.45011523601039849607669100402,
0.838290017350667771284817982326, 2.79077458676527204628834507684, 3.65227559996208901913804046244, 4.96933698777320464755573772932, 5.63458804706034937711731451563, 7.41634637329259677950715314933, 7.957156810233779731421415511348, 9.849460962713093010218892277983, 10.51080023633957191715085740985, 11.18564073878095311679514685576