| L(s) = 1 | − 2-s − 4-s + 3·8-s − 4·11-s − 13-s − 16-s + 17-s − 4·19-s + 4·22-s + 26-s + 2·29-s − 8·31-s − 5·32-s − 34-s + 2·37-s + 4·38-s − 2·41-s + 4·43-s + 4·44-s + 8·47-s − 7·49-s + 52-s − 10·53-s − 2·58-s − 4·59-s + 14·61-s + 8·62-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 1.06·8-s − 1.20·11-s − 0.277·13-s − 1/4·16-s + 0.242·17-s − 0.917·19-s + 0.852·22-s + 0.196·26-s + 0.371·29-s − 1.43·31-s − 0.883·32-s − 0.171·34-s + 0.328·37-s + 0.648·38-s − 0.312·41-s + 0.609·43-s + 0.603·44-s + 1.16·47-s − 49-s + 0.138·52-s − 1.37·53-s − 0.262·58-s − 0.520·59-s + 1.79·61-s + 1.01·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49725 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−14.67562060585128, −14.32591770450159, −13.76561484650093, −13.14554230432895, −12.72203867347717, −12.49425489968195, −11.53352664025030, −10.95117649920616, −10.60067181122836, −10.09638945562727, −9.541732397761804, −9.101696934436715, −8.461120036400101, −7.996676996333804, −7.622364957390274, −6.979727694505295, −6.324742914295962, −5.442498989655577, −5.190142937470507, −4.437803937522181, −3.907947509775914, −3.107525282147906, −2.321915387944161, −1.716265495985696, −0.7120229379600252, 0,
0.7120229379600252, 1.716265495985696, 2.321915387944161, 3.107525282147906, 3.907947509775914, 4.437803937522181, 5.190142937470507, 5.442498989655577, 6.324742914295962, 6.979727694505295, 7.622364957390274, 7.996676996333804, 8.461120036400101, 9.101696934436715, 9.541732397761804, 10.09638945562727, 10.60067181122836, 10.95117649920616, 11.53352664025030, 12.49425489968195, 12.72203867347717, 13.14554230432895, 13.76561484650093, 14.32591770450159, 14.67562060585128
