L(s) = 1 | + 2-s + 4-s + (2.12 + 0.705i)5-s + 8-s + (2.12 + 0.705i)10-s + 3.38i·11-s − 0.920·13-s + 16-s + 7.01i·17-s + 0.709i·19-s + (2.12 + 0.705i)20-s + 3.38i·22-s − 2.11·23-s + (4.00 + 2.99i)25-s − 0.920·26-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.5·4-s + (0.948 + 0.315i)5-s + 0.353·8-s + (0.671 + 0.223i)10-s + 1.01i·11-s − 0.255·13-s + 0.250·16-s + 1.70i·17-s + 0.162i·19-s + (0.474 + 0.157i)20-s + 0.720i·22-s − 0.441·23-s + (0.800 + 0.598i)25-s − 0.180·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.126 - 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.126 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.335561267\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.335561267\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-2.12 - 0.705i)T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 - 3.38iT - 11T^{2} \) |
| 13 | \( 1 + 0.920T + 13T^{2} \) |
| 17 | \( 1 - 7.01iT - 17T^{2} \) |
| 19 | \( 1 - 0.709iT - 19T^{2} \) |
| 23 | \( 1 + 2.11T + 23T^{2} \) |
| 29 | \( 1 - 0.0694iT - 29T^{2} \) |
| 31 | \( 1 + 4.09iT - 31T^{2} \) |
| 37 | \( 1 + 2.81iT - 37T^{2} \) |
| 41 | \( 1 + 10.8T + 41T^{2} \) |
| 43 | \( 1 - 7.35iT - 43T^{2} \) |
| 47 | \( 1 - 10.9iT - 47T^{2} \) |
| 53 | \( 1 - 2.79T + 53T^{2} \) |
| 59 | \( 1 - 5.80T + 59T^{2} \) |
| 61 | \( 1 - 5.18iT - 61T^{2} \) |
| 67 | \( 1 + 6.80iT - 67T^{2} \) |
| 71 | \( 1 + 13.9iT - 71T^{2} \) |
| 73 | \( 1 + 6.18T + 73T^{2} \) |
| 79 | \( 1 - 4.57T + 79T^{2} \) |
| 83 | \( 1 - 8.39iT - 83T^{2} \) |
| 89 | \( 1 + 0.636T + 89T^{2} \) |
| 97 | \( 1 + 3.00T + 97T^{2} \) |
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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
−8.412308376511543465993880244429, −7.67877770742496166099198669834, −6.86805785222493899994049783428, −6.21176027393100757455601434505, −5.68535595336673051673953600839, −4.78094189392907051907274779902, −4.08356396115471905388177189721, −3.13478163206933284818084455679, −2.13762517709946004716555142836, −1.57225343040675776267535521939,
0.66499077919999818777458415335, 1.90296647086378331354444032427, 2.77755278196965944517289182994, 3.51403057599684308548184439443, 4.64018573997670079777227161485, 5.32177884320677875343219333043, 5.72031954091012837453114875081, 6.78327930062555110412503959697, 7.09203973051717333232103737707, 8.368316101278729236558753646813