L(s) = 1 | + (0.809 + 0.587i)2-s + (0.5 + 1.53i)3-s + (0.309 + 0.951i)4-s + (−0.5 + 1.53i)6-s − 0.618·7-s + (−0.309 + 0.951i)8-s + (−1.30 + 0.951i)9-s + (−1.30 + 0.951i)12-s + (−0.500 − 0.363i)14-s + (−0.809 + 0.587i)16-s − 1.61·18-s + (−0.309 − 0.951i)21-s + (0.5 + 0.363i)23-s − 1.61·24-s + (−0.809 − 0.587i)27-s + (−0.190 − 0.587i)28-s + ⋯ |
L(s) = 1 | + (0.809 + 0.587i)2-s + (0.5 + 1.53i)3-s + (0.309 + 0.951i)4-s + (−0.5 + 1.53i)6-s − 0.618·7-s + (−0.309 + 0.951i)8-s + (−1.30 + 0.951i)9-s + (−1.30 + 0.951i)12-s + (−0.500 − 0.363i)14-s + (−0.809 + 0.587i)16-s − 1.61·18-s + (−0.309 − 0.951i)21-s + (0.5 + 0.363i)23-s − 1.61·24-s + (−0.809 − 0.587i)27-s + (−0.190 − 0.587i)28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 - 0.125i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 - 0.125i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.977445347\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.977445347\) |
\(L(1)\) |
|
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 | \( 1 \) |
good | 3 | \( 1 + (-0.5 - 1.53i)T + (-0.809 + 0.587i)T^{2} \) |
| 7 | \( 1 + 0.618T + T^{2} \) |
| 11 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 13 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 17 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 19 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 23 | \( 1 + (-0.5 - 0.363i)T + (0.309 + 0.951i)T^{2} \) |
| 29 | \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \) |
| 31 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 37 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 41 | \( 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2} \) |
| 43 | \( 1 - 1.61T + T^{2} \) |
| 47 | \( 1 + (-0.5 - 1.53i)T + (-0.809 + 0.587i)T^{2} \) |
| 53 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 59 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \) |
| 67 | \( 1 + (0.618 - 1.90i)T + (-0.809 - 0.587i)T^{2} \) |
| 71 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 73 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 79 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 83 | \( 1 + (0.190 - 0.587i)T + (-0.809 - 0.587i)T^{2} \) |
| 89 | \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \) |
| 97 | \( 1 + (0.809 - 0.587i)T^{2} \) |
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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
−9.337185565426151960265831030541, −8.859463013026715928483031552976, −7.898091544576949359452345339648, −7.21245284959818266361017934380, −6.04249410243797351577465301196, −5.57457571373889726278445499898, −4.46203862343632856159862758270, −4.09924138041649695139441526880, −3.18877364614556452628517955038, −2.49823782775422098636233583311,
0.960225447932536745197182587749, 2.00617053881978415967273458275, 2.84018867329524142305940854648, 3.55155787620817657507054437706, 4.74010947108876759133897983603, 5.79360395593900857721083594973, 6.42213669012807378572691820264, 7.09868666405295316275535525753, 7.71601474248302653145730681110, 8.849213853674028492945501352622