L(s) = 1 | − 2-s + 3-s + 4-s − 2·5-s − 6-s − 8-s + 9-s + 2·10-s + 4·11-s + 12-s − 2·15-s + 16-s − 2·17-s − 18-s − 4·19-s − 2·20-s − 4·22-s + 8·23-s − 24-s − 25-s + 27-s − 2·29-s + 2·30-s − 32-s + 4·33-s + 2·34-s + 36-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.894·5-s − 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.632·10-s + 1.20·11-s + 0.288·12-s − 0.516·15-s + 1/4·16-s − 0.485·17-s − 0.235·18-s − 0.917·19-s − 0.447·20-s − 0.852·22-s + 1.66·23-s − 0.204·24-s − 1/5·25-s + 0.192·27-s − 0.371·29-s + 0.365·30-s − 0.176·32-s + 0.696·33-s + 0.342·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49686 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49686 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 13 | \( 1 \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 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
−14.89689257447759, −14.60804360750111, −13.67902255107935, −13.30175218021843, −12.66612659180850, −12.14919414599169, −11.55071874066316, −11.20593760803975, −10.73043425100018, −9.982508537689315, −9.458155846596434, −8.963809838849716, −8.549635726555560, −8.090810768310890, −7.418965586765403, −6.956943618398734, −6.522496011142769, −5.836797834852262, −4.896009542407697, −4.257514588815613, −3.811097746176673, −3.124376637724452, −2.461590401108209, −1.644603332126544, −0.9638600286739237, 0,
0.9638600286739237, 1.644603332126544, 2.461590401108209, 3.124376637724452, 3.811097746176673, 4.257514588815613, 4.896009542407697, 5.836797834852262, 6.522496011142769, 6.956943618398734, 7.418965586765403, 8.090810768310890, 8.549635726555560, 8.963809838849716, 9.458155846596434, 9.982508537689315, 10.73043425100018, 11.20593760803975, 11.55071874066316, 12.14919414599169, 12.66612659180850, 13.30175218021843, 13.67902255107935, 14.60804360750111, 14.89689257447759