| L(s) = 1 | − 2·2-s − 8·3-s + 4·4-s + 16·6-s − 7·7-s − 8·8-s + 37·9-s − 7·11-s − 32·12-s − 26·13-s + 14·14-s + 16·16-s + 44·17-s − 74·18-s + 142·19-s + 56·21-s + 14·22-s + 115·23-s + 64·24-s + 52·26-s − 80·27-s − 28·28-s + 29-s + 6·31-s − 32·32-s + 56·33-s − 88·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.53·3-s + 1/2·4-s + 1.08·6-s − 0.377·7-s − 0.353·8-s + 1.37·9-s − 0.191·11-s − 0.769·12-s − 0.554·13-s + 0.267·14-s + 1/4·16-s + 0.627·17-s − 0.968·18-s + 1.71·19-s + 0.581·21-s + 0.135·22-s + 1.04·23-s + 0.544·24-s + 0.392·26-s − 0.570·27-s − 0.188·28-s + 0.00640·29-s + 0.0347·31-s − 0.176·32-s + 0.295·33-s − 0.443·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + p T \) |
| good | 3 | \( 1 + 8 T + p^{3} T^{2} \) |
| 11 | \( 1 + 7 T + p^{3} T^{2} \) |
| 13 | \( 1 + 2 p T + p^{3} T^{2} \) |
| 17 | \( 1 - 44 T + p^{3} T^{2} \) |
| 19 | \( 1 - 142 T + p^{3} T^{2} \) |
| 23 | \( 1 - 5 p T + p^{3} T^{2} \) |
| 29 | \( 1 - T + p^{3} T^{2} \) |
| 31 | \( 1 - 6 T + p^{3} T^{2} \) |
| 37 | \( 1 + 411 T + p^{3} T^{2} \) |
| 41 | \( 1 + 444 T + p^{3} T^{2} \) |
| 43 | \( 1 - 221 T + p^{3} T^{2} \) |
| 47 | \( 1 + 258 T + p^{3} T^{2} \) |
| 53 | \( 1 - 626 T + p^{3} T^{2} \) |
| 59 | \( 1 + 162 T + p^{3} T^{2} \) |
| 61 | \( 1 + 820 T + p^{3} T^{2} \) |
| 67 | \( 1 - 519 T + p^{3} T^{2} \) |
| 71 | \( 1 - 61 T + p^{3} T^{2} \) |
| 73 | \( 1 + 1160 T + p^{3} T^{2} \) |
| 79 | \( 1 + 809 T + p^{3} T^{2} \) |
| 83 | \( 1 + 678 T + p^{3} T^{2} \) |
| 89 | \( 1 - 370 T + p^{3} T^{2} \) |
| 97 | \( 1 + 310 T + p^{3} T^{2} \) |
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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
−10.47462223347984876351738310175, −9.982569749571106923571099916602, −8.900589477821001559166906580154, −7.46660345136927215820342189682, −6.84532634497012023137327459859, −5.67529428762004644586680703484, −4.99849997692254907807528586578, −3.18471766664365620155519780883, −1.23886916222204927181806399034, 0,
1.23886916222204927181806399034, 3.18471766664365620155519780883, 4.99849997692254907807528586578, 5.67529428762004644586680703484, 6.84532634497012023137327459859, 7.46660345136927215820342189682, 8.900589477821001559166906580154, 9.982569749571106923571099916602, 10.47462223347984876351738310175
