L(s) = 1 | + (−0.809 + 0.587i)2-s + (0.309 − 0.951i)4-s + (0.309 + 0.951i)5-s + (0.309 + 0.951i)8-s + (−0.809 − 0.587i)9-s + (−0.809 − 0.587i)10-s + (−0.5 − 0.363i)13-s + (−0.809 − 0.587i)16-s + (−0.5 − 1.53i)17-s + 18-s + 20-s + (−0.809 + 0.587i)25-s + 0.618·26-s + (−0.5 + 1.53i)29-s + 32-s + ⋯ |
L(s) = 1 | + (−0.809 + 0.587i)2-s + (0.309 − 0.951i)4-s + (0.309 + 0.951i)5-s + (0.309 + 0.951i)8-s + (−0.809 − 0.587i)9-s + (−0.809 − 0.587i)10-s + (−0.5 − 0.363i)13-s + (−0.809 − 0.587i)16-s + (−0.5 − 1.53i)17-s + 18-s + 20-s + (−0.809 + 0.587i)25-s + 0.618·26-s + (−0.5 + 1.53i)29-s + 32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.728 - 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.728 - 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4075006026\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4075006026\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.809 - 0.587i)T \) |
| 5 | \( 1 + (-0.309 - 0.951i)T \) |
good | 3 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 13 | \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \) |
| 17 | \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \) |
| 19 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 23 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 29 | \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \) |
| 31 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 37 | \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \) |
| 41 | \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 53 | \( 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2} \) |
| 59 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 61 | \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \) |
| 67 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 71 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 73 | \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \) |
| 79 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 83 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 89 | \( 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2} \) |
| 97 | \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \) |
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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
−14.54597469732440986161986399890, −13.61194365105627545176539506433, −11.79533847394713679437616359153, −10.91836552352720941025338915605, −9.806097530092695307679460379295, −8.900279530223792754793762110576, −7.46697379647670625499317743355, −6.55853129117874277206299100115, −5.32983035558480357806833535857, −2.77583471385176013367479650464,
2.16566075758153450204676947752, 4.26746565859084420235790164286, 5.99243829011274036691812126531, 7.79299994935581846419386074728, 8.655325615439447738633677550011, 9.621676144300964856013990338315, 10.78338008992297043504455948009, 11.80942225746427584641422958691, 12.80964274905523418157293362557, 13.66748595185211050043441405446