L(s) = 1 | + (0.866 + 0.5i)2-s − i·8-s + (0.5 − 0.866i)9-s + (0.5 + 0.866i)11-s + (0.5 − 0.866i)16-s + (0.866 − 0.499i)18-s + 0.999i·22-s + (−0.866 − 0.5i)23-s + 29-s + (0.866 + 0.5i)37-s + i·43-s + (−0.499 − 0.866i)46-s + (−1.73 + i)53-s + (0.866 + 0.5i)58-s − 64-s + ⋯ |
L(s) = 1 | + (0.866 + 0.5i)2-s − i·8-s + (0.5 − 0.866i)9-s + (0.5 + 0.866i)11-s + (0.5 − 0.866i)16-s + (0.866 − 0.499i)18-s + 0.999i·22-s + (−0.866 − 0.5i)23-s + 29-s + (0.866 + 0.5i)37-s + i·43-s + (−0.499 − 0.866i)46-s + (−1.73 + i)53-s + (0.866 + 0.5i)58-s − 64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.00342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.00342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.630266111\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.630266111\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 3 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 - T + T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - iT - T^{2} \) |
| 47 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (1.73 - i)T + (0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + T + T^{2} \) |
| 73 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + T^{2} \) |
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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
−9.757164784518879194895188463418, −9.354242340943961260233146548285, −8.144329087936616228688256329159, −7.11801742361243060241271858594, −6.48433447341575731455902893978, −5.84666717386432890729971227593, −4.54118539395253706201611134673, −4.26322602047823058519856745019, −3.03678123928940308422767922130, −1.34952014963397499355087515702,
1.76351664832382944833496101964, 2.89538604763926493334569731822, 3.85225095112372907557771622793, 4.62448274509125449747838576721, 5.50593620852497924811778099128, 6.37205653261201891536893642438, 7.59845686118879749840970484657, 8.229691114857010938940363562602, 9.077430605455813115827737050822, 10.13329446623597806298932814914