L(s) = 1 | − 2.77e4i·2-s + 6.45e7i·3-s + 7.82e9·4-s + (−2.89e11 + 1.80e11i)5-s + 1.78e12·6-s − 3.09e13i·7-s − 4.55e14i·8-s + 1.39e15·9-s + (5.01e15 + 8.01e15i)10-s + 1.96e17·11-s + 5.04e17i·12-s − 4.53e18i·13-s − 8.57e17·14-s + (−1.16e19 − 1.86e19i)15-s + 5.45e19·16-s + 2.94e20i·17-s + ⋯ |
L(s) = 1 | − 0.299i·2-s + 0.865i·3-s + 0.910·4-s + (−0.847 + 0.530i)5-s + 0.258·6-s − 0.351i·7-s − 0.571i·8-s + 0.251·9-s + (0.158 + 0.253i)10-s + 1.28·11-s + 0.787i·12-s − 1.88i·13-s − 0.105·14-s + (−0.458 − 0.733i)15-s + 0.739·16-s + 1.46i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.847 - 0.530i)\, \overline{\Lambda}(34-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+33/2) \, L(s)\cr =\mathstrut & (0.847 - 0.530i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(17)\) |
\(\approx\) |
\(2.525797979\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.525797979\) |
\(L(\frac{35}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (2.89e11 - 1.80e11i)T \) |
good | 2 | \( 1 + 2.77e4iT - 8.58e9T^{2} \) |
| 3 | \( 1 - 6.45e7iT - 5.55e15T^{2} \) |
| 7 | \( 1 + 3.09e13iT - 7.73e27T^{2} \) |
| 11 | \( 1 - 1.96e17T + 2.32e34T^{2} \) |
| 13 | \( 1 + 4.53e18iT - 5.75e36T^{2} \) |
| 17 | \( 1 - 2.94e20iT - 4.02e40T^{2} \) |
| 19 | \( 1 + 6.15e20T + 1.58e42T^{2} \) |
| 23 | \( 1 - 1.65e22iT - 8.65e44T^{2} \) |
| 29 | \( 1 - 9.17e22T + 1.81e48T^{2} \) |
| 31 | \( 1 - 5.75e24T + 1.64e49T^{2} \) |
| 37 | \( 1 - 9.32e25iT - 5.63e51T^{2} \) |
| 41 | \( 1 - 4.59e26T + 1.66e53T^{2} \) |
| 43 | \( 1 - 2.98e26iT - 8.02e53T^{2} \) |
| 47 | \( 1 - 3.83e27iT - 1.51e55T^{2} \) |
| 53 | \( 1 + 1.34e28iT - 7.96e56T^{2} \) |
| 59 | \( 1 + 1.39e29T + 2.74e58T^{2} \) |
| 61 | \( 1 + 1.27e29T + 8.23e58T^{2} \) |
| 67 | \( 1 + 1.55e30iT - 1.82e60T^{2} \) |
| 71 | \( 1 - 4.80e30T + 1.23e61T^{2} \) |
| 73 | \( 1 - 2.54e30iT - 3.08e61T^{2} \) |
| 79 | \( 1 + 1.90e31T + 4.18e62T^{2} \) |
| 83 | \( 1 - 1.69e31iT - 2.13e63T^{2} \) |
| 89 | \( 1 - 8.93e31T + 2.13e64T^{2} \) |
| 97 | \( 1 - 6.99e32iT - 3.65e65T^{2} \) |
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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
−15.59044174624463283934122871130, −14.96915391222112998203337347452, −12.46876022711655236984423784336, −10.98676575311603066251101367964, −10.15126609951797230971997197440, −7.900064662779545501849474259975, −6.36856823467558834474314239060, −4.09978249318805670901826062251, −3.14725876519792720652581923406, −1.11205112994563700428511931981,
0.962016027512592593022638201663, 2.19417139513139080618556006359, 4.30393660887613456647790715850, 6.50287378062508941756587404473, 7.33162527604516630437690261749, 8.970552929463708667807567272157, 11.59107087599806358292061639173, 12.17377331837217645847362272287, 14.18380884334344382236437586608, 15.81630247477563510575180421466