L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 2·7-s + 8-s + 9-s − 11-s + 12-s + 4·13-s + 2·14-s + 16-s + 6·17-s + 18-s + 4·19-s + 2·21-s − 22-s − 6·23-s + 24-s − 5·25-s + 4·26-s + 27-s + 2·28-s − 6·29-s − 8·31-s + 32-s − 33-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.755·7-s + 0.353·8-s + 1/3·9-s − 0.301·11-s + 0.288·12-s + 1.10·13-s + 0.534·14-s + 1/4·16-s + 1.45·17-s + 0.235·18-s + 0.917·19-s + 0.436·21-s − 0.213·22-s − 1.25·23-s + 0.204·24-s − 25-s + 0.784·26-s + 0.192·27-s + 0.377·28-s − 1.11·29-s − 1.43·31-s + 0.176·32-s − 0.174·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 90354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90354 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 37 | \( 1 \) |
good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−14.02889940689037, −13.77584213978275, −13.15295255634067, −12.81721560118132, −12.08546191268037, −11.77275787404324, −11.14780917994328, −10.87625094355762, −10.02563002538052, −9.752747409421075, −9.161779934290534, −8.337542331422547, −8.068505503264153, −7.525587277916313, −7.229152572617807, −6.252584330450435, −5.778414489637384, −5.424634261340309, −4.791811316210028, −4.000653093651710, −3.632039864502596, −3.199248978072920, −2.369906236722728, −1.509550597653004, −1.447237668144024, 0,
1.447237668144024, 1.509550597653004, 2.369906236722728, 3.199248978072920, 3.632039864502596, 4.000653093651710, 4.791811316210028, 5.424634261340309, 5.778414489637384, 6.252584330450435, 7.229152572617807, 7.525587277916313, 8.068505503264153, 8.337542331422547, 9.161779934290534, 9.752747409421075, 10.02563002538052, 10.87625094355762, 11.14780917994328, 11.77275787404324, 12.08546191268037, 12.81721560118132, 13.15295255634067, 13.77584213978275, 14.02889940689037