L(s) = 1 | + (0.348 + 0.137i)3-s + (0.535 + 0.844i)4-s + (0.309 − 0.951i)5-s + (−0.626 − 0.588i)9-s + (0.876 + 0.481i)11-s + (0.0702 + 0.368i)12-s + (0.238 − 0.288i)15-s + (−0.425 + 0.904i)16-s + (0.968 − 0.248i)20-s + (1.84 − 0.233i)23-s + (−0.809 − 0.587i)25-s + (−0.296 − 0.630i)27-s + (−1.06 + 1.67i)31-s + (0.238 + 0.288i)33-s + (0.161 − 0.844i)36-s + (0.688 − 1.46i)37-s + ⋯ |
L(s) = 1 | + (0.348 + 0.137i)3-s + (0.535 + 0.844i)4-s + (0.309 − 0.951i)5-s + (−0.626 − 0.588i)9-s + (0.876 + 0.481i)11-s + (0.0702 + 0.368i)12-s + (0.238 − 0.288i)15-s + (−0.425 + 0.904i)16-s + (0.968 − 0.248i)20-s + (1.84 − 0.233i)23-s + (−0.809 − 0.587i)25-s + (−0.296 − 0.630i)27-s + (−1.06 + 1.67i)31-s + (0.238 + 0.288i)33-s + (0.161 − 0.844i)36-s + (0.688 − 1.46i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 - 0.150i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 - 0.150i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.435167614\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.435167614\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-0.309 + 0.951i)T \) |
| 11 | \( 1 + (-0.876 - 0.481i)T \) |
good | 2 | \( 1 + (-0.535 - 0.844i)T^{2} \) |
| 3 | \( 1 + (-0.348 - 0.137i)T + (0.728 + 0.684i)T^{2} \) |
| 7 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 13 | \( 1 + (-0.0627 - 0.998i)T^{2} \) |
| 17 | \( 1 + (0.425 + 0.904i)T^{2} \) |
| 19 | \( 1 + (-0.728 + 0.684i)T^{2} \) |
| 23 | \( 1 + (-1.84 + 0.233i)T + (0.968 - 0.248i)T^{2} \) |
| 29 | \( 1 + (-0.876 + 0.481i)T^{2} \) |
| 31 | \( 1 + (1.06 - 1.67i)T + (-0.425 - 0.904i)T^{2} \) |
| 37 | \( 1 + (-0.688 + 1.46i)T + (-0.637 - 0.770i)T^{2} \) |
| 41 | \( 1 + (-0.968 - 0.248i)T^{2} \) |
| 43 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 47 | \( 1 + (-0.0388 + 0.616i)T + (-0.992 - 0.125i)T^{2} \) |
| 53 | \( 1 + (1.11 - 1.35i)T + (-0.187 - 0.982i)T^{2} \) |
| 59 | \( 1 + (-0.159 - 0.836i)T + (-0.929 + 0.368i)T^{2} \) |
| 61 | \( 1 + (-0.968 + 0.248i)T^{2} \) |
| 67 | \( 1 + (1.23 + 0.317i)T + (0.876 + 0.481i)T^{2} \) |
| 71 | \( 1 + (0.0235 - 0.374i)T + (-0.992 - 0.125i)T^{2} \) |
| 73 | \( 1 + (0.929 + 0.368i)T^{2} \) |
| 79 | \( 1 + (-0.728 - 0.684i)T^{2} \) |
| 83 | \( 1 + (-0.728 + 0.684i)T^{2} \) |
| 89 | \( 1 + (0.0235 - 0.123i)T + (-0.929 - 0.368i)T^{2} \) |
| 97 | \( 1 + (0.824 - 0.211i)T + (0.876 - 0.481i)T^{2} \) |
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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
−9.362348453635640077860947291150, −9.026699974090399570028341975393, −8.404048952552510027870866011393, −7.35797345478071954870516970670, −6.65692067327776830410433213204, −5.67063426912688996027434976286, −4.59954289946479893004043332245, −3.68814299165976853798798977122, −2.78497824750466030228302951974, −1.50653561665538077210631884571,
1.53408678243934952776277110015, 2.61236435278439496055815234491, 3.36923559371122185973073482625, 4.87366628280298513786281417357, 5.82631661314716290250808884704, 6.44057779087443568833325492337, 7.21339590637989932253254989073, 8.046314869775991540754831587857, 9.202981874719485476734629644551, 9.660069335815935025392222887233