L(s) = 1 | + (−0.707 + 0.707i)3-s + (1.68 + 1.47i)5-s + (2.29 − 2.29i)7-s − 1.00i·9-s + 5.35i·11-s + (−2.00 + 2.00i)13-s + (−2.23 + 0.151i)15-s + (−3.44 + 3.44i)17-s + 0.00250·19-s + 3.24i·21-s + (−4.38 − 1.94i)23-s + (0.676 + 4.95i)25-s + (0.707 + 0.707i)27-s + 1.32i·29-s + 7.62·31-s + ⋯ |
L(s) = 1 | + (−0.408 + 0.408i)3-s + (0.753 + 0.657i)5-s + (0.867 − 0.867i)7-s − 0.333i·9-s + 1.61i·11-s + (−0.556 + 0.556i)13-s + (−0.576 + 0.0391i)15-s + (−0.836 + 0.836i)17-s + 0.000575·19-s + 0.708i·21-s + (−0.913 − 0.405i)23-s + (0.135 + 0.990i)25-s + (0.136 + 0.136i)27-s + 0.245i·29-s + 1.36·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.202 - 0.979i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.202 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.537827391\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.537827391\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.707 - 0.707i)T \) |
| 5 | \( 1 + (-1.68 - 1.47i)T \) |
| 23 | \( 1 + (4.38 + 1.94i)T \) |
good | 7 | \( 1 + (-2.29 + 2.29i)T - 7iT^{2} \) |
| 11 | \( 1 - 5.35iT - 11T^{2} \) |
| 13 | \( 1 + (2.00 - 2.00i)T - 13iT^{2} \) |
| 17 | \( 1 + (3.44 - 3.44i)T - 17iT^{2} \) |
| 19 | \( 1 - 0.00250T + 19T^{2} \) |
| 29 | \( 1 - 1.32iT - 29T^{2} \) |
| 31 | \( 1 - 7.62T + 31T^{2} \) |
| 37 | \( 1 + (-6.26 + 6.26i)T - 37iT^{2} \) |
| 41 | \( 1 + 7.53T + 41T^{2} \) |
| 43 | \( 1 + (-0.265 - 0.265i)T + 43iT^{2} \) |
| 47 | \( 1 + (-8.13 - 8.13i)T + 47iT^{2} \) |
| 53 | \( 1 + (2.78 + 2.78i)T + 53iT^{2} \) |
| 59 | \( 1 - 10.7iT - 59T^{2} \) |
| 61 | \( 1 - 2.04iT - 61T^{2} \) |
| 67 | \( 1 + (2.91 - 2.91i)T - 67iT^{2} \) |
| 71 | \( 1 - 7.55T + 71T^{2} \) |
| 73 | \( 1 + (0.944 - 0.944i)T - 73iT^{2} \) |
| 79 | \( 1 + 6.43T + 79T^{2} \) |
| 83 | \( 1 + (-2.53 - 2.53i)T + 83iT^{2} \) |
| 89 | \( 1 + 8.05T + 89T^{2} \) |
| 97 | \( 1 + (9.47 - 9.47i)T - 97iT^{2} \) |
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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
−9.970586191057899283013335858461, −9.280134114546224133170457814228, −8.098593601132019694286137175690, −7.18959875224903449081355159576, −6.64528006827563070148964094507, −5.66100304625340804300506243817, −4.43252677116888987096171033450, −4.28833041486067060925546588454, −2.49952431089934730962157787160, −1.60370389875084646876650019820,
0.65734977524275073441755692905, 1.96787764787235624858678744847, 2.90023212106959736401777134582, 4.56109445426932645002877132052, 5.33079676358966026153988082483, 5.87406127947815624908292018666, 6.69671092094547206402136092903, 8.122671000701965868209381277504, 8.331849212552356297019234606229, 9.278206657615928914574771317182