L(s) = 1 | + (0.562 + 0.562i)2-s + (0.534 − 0.534i)3-s − 1.36i·4-s + (0.233 + 2.22i)5-s + 0.601·6-s + (0.567 − 0.567i)7-s + (1.89 − 1.89i)8-s + 2.42i·9-s + (−1.11 + 1.38i)10-s − 4.83i·11-s + (−0.730 − 0.730i)12-s + (−3.26 + 3.26i)13-s + 0.638·14-s + (1.31 + 1.06i)15-s − 0.601·16-s + (−1.54 + 1.54i)17-s + ⋯ |
L(s) = 1 | + (0.397 + 0.397i)2-s + (0.308 − 0.308i)3-s − 0.683i·4-s + (0.104 + 0.994i)5-s + 0.245·6-s + (0.214 − 0.214i)7-s + (0.669 − 0.669i)8-s + 0.809i·9-s + (−0.354 + 0.437i)10-s − 1.45i·11-s + (−0.210 − 0.210i)12-s + (−0.904 + 0.904i)13-s + 0.170·14-s + (0.339 + 0.274i)15-s − 0.150·16-s + (−0.375 + 0.375i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.120i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 - 0.120i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.36643 + 0.0826235i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.36643 + 0.0826235i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-0.233 - 2.22i)T \) |
| 23 | \( 1 + (-4.11 - 2.45i)T \) |
good | 2 | \( 1 + (-0.562 - 0.562i)T + 2iT^{2} \) |
| 3 | \( 1 + (-0.534 + 0.534i)T - 3iT^{2} \) |
| 7 | \( 1 + (-0.567 + 0.567i)T - 7iT^{2} \) |
| 11 | \( 1 + 4.83iT - 11T^{2} \) |
| 13 | \( 1 + (3.26 - 3.26i)T - 13iT^{2} \) |
| 17 | \( 1 + (1.54 - 1.54i)T - 17iT^{2} \) |
| 19 | \( 1 + 7.21T + 19T^{2} \) |
| 29 | \( 1 + 5.91iT - 29T^{2} \) |
| 31 | \( 1 + 1.58T + 31T^{2} \) |
| 37 | \( 1 + (-6.03 + 6.03i)T - 37iT^{2} \) |
| 41 | \( 1 - 1.73T + 41T^{2} \) |
| 43 | \( 1 + (-4.49 - 4.49i)T + 43iT^{2} \) |
| 47 | \( 1 + (-0.427 - 0.427i)T + 47iT^{2} \) |
| 53 | \( 1 + (-6.85 - 6.85i)T + 53iT^{2} \) |
| 59 | \( 1 + 7.76iT - 59T^{2} \) |
| 61 | \( 1 + 7.62iT - 61T^{2} \) |
| 67 | \( 1 + (1.20 - 1.20i)T - 67iT^{2} \) |
| 71 | \( 1 - 10.2T + 71T^{2} \) |
| 73 | \( 1 + (-0.559 + 0.559i)T - 73iT^{2} \) |
| 79 | \( 1 + 5.66T + 79T^{2} \) |
| 83 | \( 1 + (-2.15 - 2.15i)T + 83iT^{2} \) |
| 89 | \( 1 - 9.48T + 89T^{2} \) |
| 97 | \( 1 + (5.03 - 5.03i)T - 97iT^{2} \) |
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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
−13.87656893333041018718809908157, −13.01758902468304911317570711354, −11.10174307602582834663075116027, −10.78336766175849065026819683256, −9.366503132806725689116501323855, −7.895299436328184395907906534887, −6.80127114110312599237660363162, −5.84568366326609710686362661003, −4.29678248344262617905171437257, −2.29820057851878747187405259878,
2.44373488218025080121320440329, 4.18523488298605658139856115696, 5.02060918265720446050667848976, 7.04076997035707485814186688321, 8.346877195762357597875704511573, 9.200795823388528940857347465243, 10.39280601761871692130484285709, 11.90827706960177346766686308433, 12.59733440452461084473775045031, 13.08977255955906051698654851957