L(s) = 1 | + (0.453 + 0.891i)2-s + (−0.587 + 0.809i)4-s + (1.59 − 0.253i)7-s + (−0.987 − 0.156i)8-s + (−0.951 − 0.309i)9-s + (0.809 + 0.587i)11-s + (0.550 − 0.280i)13-s + (0.951 + 1.30i)14-s + (−0.309 − 0.951i)16-s + (−0.156 − 0.987i)18-s + (1.53 − 1.11i)19-s + (−0.156 + 0.987i)22-s + (−0.831 + 0.831i)23-s + (0.5 + 0.363i)26-s + (−0.734 + 1.44i)28-s + ⋯ |
L(s) = 1 | + (0.453 + 0.891i)2-s + (−0.587 + 0.809i)4-s + (1.59 − 0.253i)7-s + (−0.987 − 0.156i)8-s + (−0.951 − 0.309i)9-s + (0.809 + 0.587i)11-s + (0.550 − 0.280i)13-s + (0.951 + 1.30i)14-s + (−0.309 − 0.951i)16-s + (−0.156 − 0.987i)18-s + (1.53 − 1.11i)19-s + (−0.156 + 0.987i)22-s + (−0.831 + 0.831i)23-s + (0.5 + 0.363i)26-s + (−0.734 + 1.44i)28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.251 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.251 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.629540776\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.629540776\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.453 - 0.891i)T \) |
| 5 | \( 1 \) |
| 11 | \( 1 + (-0.809 - 0.587i)T \) |
good | 3 | \( 1 + (0.951 + 0.309i)T^{2} \) |
| 7 | \( 1 + (-1.59 + 0.253i)T + (0.951 - 0.309i)T^{2} \) |
| 13 | \( 1 + (-0.550 + 0.280i)T + (0.587 - 0.809i)T^{2} \) |
| 17 | \( 1 + (-0.587 - 0.809i)T^{2} \) |
| 19 | \( 1 + (-1.53 + 1.11i)T + (0.309 - 0.951i)T^{2} \) |
| 23 | \( 1 + (0.831 - 0.831i)T - iT^{2} \) |
| 29 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 31 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 37 | \( 1 + (-0.297 - 1.87i)T + (-0.951 + 0.309i)T^{2} \) |
| 41 | \( 1 + (0.690 + 0.951i)T + (-0.309 + 0.951i)T^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 47 | \( 1 + (1.87 + 0.297i)T + (0.951 + 0.309i)T^{2} \) |
| 53 | \( 1 + (-0.533 - 1.04i)T + (-0.587 + 0.809i)T^{2} \) |
| 59 | \( 1 + (0.363 - 0.5i)T + (-0.309 - 0.951i)T^{2} \) |
| 61 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 67 | \( 1 - iT^{2} \) |
| 71 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 73 | \( 1 + (0.951 - 0.309i)T^{2} \) |
| 79 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 83 | \( 1 + (0.587 + 0.809i)T^{2} \) |
| 89 | \( 1 + 1.61iT - T^{2} \) |
| 97 | \( 1 + (-0.587 + 0.809i)T^{2} \) |
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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
−9.108992815004614879270124854973, −8.447722452197595893902652909145, −7.81777777969285629598417511824, −7.12545643152932192307201549858, −6.24867603902400742700681024722, −5.34224038604375208752307604847, −4.83320206717961482693720526421, −3.88208207676307091802129895328, −2.95618074618315093804851350559, −1.39767260230612555240245918446,
1.28402221717996377238825219912, 2.12417116422083869887007920473, 3.30566633401751964670236849430, 4.10065613702349084832549392451, 5.09374738055191917319604718970, 5.64245301266492908376476004610, 6.42383995604190590431074089109, 7.936832562273825944824872539706, 8.335577383055783370802806658281, 9.101721188055335795330841337907