L(s) = 1 | − 4.94·3-s + (−3.69 + 3.37i)5-s + (3.22 + 3.22i)7-s + 15.4·9-s + (−7.67 − 7.67i)11-s + 22.2·13-s + (18.2 − 16.6i)15-s + (−3.76 + 3.76i)17-s + (0.809 + 0.809i)19-s + (−15.9 − 15.9i)21-s + (12.2 − 12.2i)23-s + (2.23 − 24.8i)25-s − 31.8·27-s + (27.1 + 27.1i)29-s − 25.5·31-s + ⋯ |
L(s) = 1 | − 1.64·3-s + (−0.738 + 0.674i)5-s + (0.460 + 0.460i)7-s + 1.71·9-s + (−0.697 − 0.697i)11-s + 1.70·13-s + (1.21 − 1.11i)15-s + (−0.221 + 0.221i)17-s + (0.0426 + 0.0426i)19-s + (−0.759 − 0.759i)21-s + (0.534 − 0.534i)23-s + (0.0895 − 0.995i)25-s − 1.18·27-s + (0.936 + 0.936i)29-s − 0.824·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.587 - 0.809i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.587 - 0.809i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.5603033335\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5603033335\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (3.69 - 3.37i)T \) |
good | 3 | \( 1 + 4.94T + 9T^{2} \) |
| 7 | \( 1 + (-3.22 - 3.22i)T + 49iT^{2} \) |
| 11 | \( 1 + (7.67 + 7.67i)T + 121iT^{2} \) |
| 13 | \( 1 - 22.2T + 169T^{2} \) |
| 17 | \( 1 + (3.76 - 3.76i)T - 289iT^{2} \) |
| 19 | \( 1 + (-0.809 - 0.809i)T + 361iT^{2} \) |
| 23 | \( 1 + (-12.2 + 12.2i)T - 529iT^{2} \) |
| 29 | \( 1 + (-27.1 - 27.1i)T + 841iT^{2} \) |
| 31 | \( 1 + 25.5T + 961T^{2} \) |
| 37 | \( 1 + 8.62T + 1.36e3T^{2} \) |
| 41 | \( 1 - 73.4iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 17.5iT - 1.84e3T^{2} \) |
| 47 | \( 1 + (17.4 - 17.4i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 - 29.3iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (72.6 - 72.6i)T - 3.48e3iT^{2} \) |
| 61 | \( 1 + (19.7 - 19.7i)T - 3.72e3iT^{2} \) |
| 67 | \( 1 + 14.0iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 91.3iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (-42.7 + 42.7i)T - 5.32e3iT^{2} \) |
| 79 | \( 1 - 46.7iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 82.9T + 6.88e3T^{2} \) |
| 89 | \( 1 + 131.T + 7.92e3T^{2} \) |
| 97 | \( 1 + (30.5 - 30.5i)T - 9.40e3iT^{2} \) |
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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
−10.96753698134813063373861101852, −10.35858188152178206794156169639, −8.774845610836210518090179698140, −8.020936590534934339826998634193, −6.82710512435545569571382714619, −6.16387268378203888075931934006, −5.35397146082264605203697947750, −4.33789186454049034223005304787, −3.09774753728106826111379165548, −1.13898290397544926325191972372,
0.32546829705407005057530735336, 1.44195150816041835858200229243, 3.76638187411828792181222444679, 4.69679815759383479972594869019, 5.35271661485362289713354598529, 6.37276508572491515737662004115, 7.33567832541661782646884594726, 8.157317481260078207439140177489, 9.261952143216883302113468253478, 10.49261598174586054504777170452