L(s) = 1 | + (0.755 + 0.654i)2-s + (0.142 + 0.989i)4-s + (0.462 + 1.38i)5-s + (−0.540 + 0.841i)8-s + (0.800 − 0.599i)9-s + (−0.560 + 1.35i)10-s + (0.0956 − 0.890i)13-s + (−0.959 + 0.281i)16-s + (0.153 − 0.239i)17-s + (0.997 + 0.0713i)18-s + (−1.30 + 0.655i)20-s + (−0.914 + 0.684i)25-s + (0.655 − 0.610i)26-s + (−1.73 − 0.249i)29-s + (−0.909 − 0.415i)32-s + ⋯ |
L(s) = 1 | + (0.755 + 0.654i)2-s + (0.142 + 0.989i)4-s + (0.462 + 1.38i)5-s + (−0.540 + 0.841i)8-s + (0.800 − 0.599i)9-s + (−0.560 + 1.35i)10-s + (0.0956 − 0.890i)13-s + (−0.959 + 0.281i)16-s + (0.153 − 0.239i)17-s + (0.997 + 0.0713i)18-s + (−1.30 + 0.655i)20-s + (−0.914 + 0.684i)25-s + (0.655 − 0.610i)26-s + (−1.73 − 0.249i)29-s + (−0.909 − 0.415i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1412 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.190 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1412 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.190 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.781728723\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.781728723\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.755 - 0.654i)T \) |
| 353 | \( 1 + (-0.841 + 0.540i)T \) |
good | 3 | \( 1 + (-0.800 + 0.599i)T^{2} \) |
| 5 | \( 1 + (-0.462 - 1.38i)T + (-0.800 + 0.599i)T^{2} \) |
| 7 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 11 | \( 1 + (-0.654 + 0.755i)T^{2} \) |
| 13 | \( 1 + (-0.0956 + 0.890i)T + (-0.977 - 0.212i)T^{2} \) |
| 17 | \( 1 + (-0.153 + 0.239i)T + (-0.415 - 0.909i)T^{2} \) |
| 19 | \( 1 + (-0.540 + 0.841i)T^{2} \) |
| 23 | \( 1 + (0.755 - 0.654i)T^{2} \) |
| 29 | \( 1 + (1.73 + 0.249i)T + (0.959 + 0.281i)T^{2} \) |
| 31 | \( 1 + (0.997 - 0.0713i)T^{2} \) |
| 37 | \( 1 + (-0.349 - 0.0630i)T + (0.936 + 0.349i)T^{2} \) |
| 41 | \( 1 + (1.57 + 0.587i)T + (0.755 + 0.654i)T^{2} \) |
| 43 | \( 1 + (-0.755 - 0.654i)T^{2} \) |
| 47 | \( 1 + (0.281 + 0.959i)T^{2} \) |
| 53 | \( 1 + (0.912 + 0.456i)T + (0.599 + 0.800i)T^{2} \) |
| 59 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 61 | \( 1 + (-0.270 - 0.919i)T + (-0.841 + 0.540i)T^{2} \) |
| 67 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 71 | \( 1 + (-0.479 + 0.877i)T^{2} \) |
| 73 | \( 1 + (-1.39 + 0.201i)T + (0.959 - 0.281i)T^{2} \) |
| 79 | \( 1 + (0.800 + 0.599i)T^{2} \) |
| 83 | \( 1 + (0.415 - 0.909i)T^{2} \) |
| 89 | \( 1 + (-0.241 - 1.34i)T + (-0.936 + 0.349i)T^{2} \) |
| 97 | \( 1 + (-1.10 + 0.708i)T + (0.415 - 0.909i)T^{2} \) |
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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
−10.02103338500347877197250335781, −9.191531179969621123700716817924, −8.030904089729238109952044990578, −7.26314949946901973030776533502, −6.72184602777809163945908885021, −5.95398572954505067299254515294, −5.15594820944063742305865321272, −3.84040862854128114947103283823, −3.25525271869992410595548148089, −2.15513692969663137716165720318,
1.39281605264563715665157177209, 2.05611397282145582788654207181, 3.64687072918391729946712308653, 4.53275751729311731176567804870, 5.08699485788676562465696750014, 5.92904217926955213217621707949, 6.91363267502492777315174242508, 7.982325968723163626189896745232, 9.086707374165632532726887980185, 9.501610668170097810069677722965