L(s) = 1 | + (8.62e3 + 8.62e3i)2-s + (9.89e7 − 9.89e7i)3-s − 1.70e10i·4-s + (−8.06e10 + 7.58e11i)5-s + 1.70e12·6-s + (1.10e14 + 1.10e14i)7-s + (2.95e14 − 2.95e14i)8-s − 2.88e15i·9-s + (−7.24e15 + 5.84e15i)10-s − 6.88e17·11-s + (−1.68e18 − 1.68e18i)12-s + (−6.48e18 + 6.48e18i)13-s + 1.91e18i·14-s + (6.70e19 + 8.30e19i)15-s − 2.87e20·16-s + (−2.93e20 − 2.93e20i)17-s + ⋯ |
L(s) = 1 | + (0.0658 + 0.0658i)2-s + (0.765 − 0.765i)3-s − 0.991i·4-s + (−0.105 + 0.994i)5-s + 0.100·6-s + (0.476 + 0.476i)7-s + (0.131 − 0.131i)8-s − 0.173i·9-s + (−0.0724 + 0.0584i)10-s − 1.36·11-s + (−0.759 − 0.759i)12-s + (−0.750 + 0.750i)13-s + 0.0627i·14-s + (0.680 + 0.842i)15-s − 0.974·16-s + (−0.354 − 0.354i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.432 - 0.901i)\, \overline{\Lambda}(35-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+17) \, L(s)\cr =\mathstrut & (-0.432 - 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{35}{2})\) |
\(\approx\) |
\(0.8282603649\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8282603649\) |
\(L(18)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (8.06e10 - 7.58e11i)T \) |
good | 2 | \( 1 + (-8.62e3 - 8.62e3i)T + 1.71e10iT^{2} \) |
| 3 | \( 1 + (-9.89e7 + 9.89e7i)T - 1.66e16iT^{2} \) |
| 7 | \( 1 + (-1.10e14 - 1.10e14i)T + 5.41e28iT^{2} \) |
| 11 | \( 1 + 6.88e17T + 2.55e35T^{2} \) |
| 13 | \( 1 + (6.48e18 - 6.48e18i)T - 7.48e37iT^{2} \) |
| 17 | \( 1 + (2.93e20 + 2.93e20i)T + 6.84e41iT^{2} \) |
| 19 | \( 1 + 1.21e21iT - 3.00e43T^{2} \) |
| 23 | \( 1 + (1.01e23 - 1.01e23i)T - 1.98e46iT^{2} \) |
| 29 | \( 1 + 1.11e23iT - 5.26e49T^{2} \) |
| 31 | \( 1 - 5.21e24T + 5.08e50T^{2} \) |
| 37 | \( 1 + (-3.43e26 - 3.43e26i)T + 2.08e53iT^{2} \) |
| 41 | \( 1 + 3.54e27T + 6.83e54T^{2} \) |
| 43 | \( 1 + (7.17e27 - 7.17e27i)T - 3.45e55iT^{2} \) |
| 47 | \( 1 + (-1.36e28 - 1.36e28i)T + 7.10e56iT^{2} \) |
| 53 | \( 1 + (2.76e28 - 2.76e28i)T - 4.22e58iT^{2} \) |
| 59 | \( 1 + 1.87e30iT - 1.61e60T^{2} \) |
| 61 | \( 1 - 2.15e30T + 5.02e60T^{2} \) |
| 67 | \( 1 + (6.36e30 + 6.36e30i)T + 1.22e62iT^{2} \) |
| 71 | \( 1 - 5.86e31T + 8.76e62T^{2} \) |
| 73 | \( 1 + (5.09e31 - 5.09e31i)T - 2.25e63iT^{2} \) |
| 79 | \( 1 - 3.07e32iT - 3.30e64T^{2} \) |
| 83 | \( 1 + (-2.32e32 + 2.32e32i)T - 1.77e65iT^{2} \) |
| 89 | \( 1 + 2.13e33iT - 1.90e66T^{2} \) |
| 97 | \( 1 + (-2.18e33 - 2.18e33i)T + 3.55e67iT^{2} \) |
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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
−15.53523906526692155797739025841, −14.46337257509999394074295231831, −13.46966954858052030709336376292, −11.37530655489444828066681076744, −9.897163796856779111816416124082, −8.009626143831810923153972130096, −6.75496803162955576226733923635, −5.06246117796117957310910835463, −2.64932531366238405213734899123, −1.84060162403662717268775705359,
0.19248735591042260209229275586, 2.42877056733720794014044835067, 3.84403109956904798396832792088, 4.93085144524915651035295224932, 7.78003455686554666620082538127, 8.609671161907087791164460563487, 10.25575571684940985391088256358, 12.26064678972042997996168653338, 13.47213553888872695662243647963, 15.25767205189090815684292199248