L(s) = 1 | − 2-s − 4-s + 3·7-s + 3·8-s − 2·11-s + 2·13-s − 3·14-s − 16-s − 4·17-s − 8·19-s + 2·22-s − 3·23-s − 2·26-s − 3·28-s − 29-s − 5·32-s + 4·34-s + 4·37-s + 8·38-s + 5·41-s + 8·43-s + 2·44-s + 3·46-s − 7·47-s + 2·49-s − 2·52-s + 2·53-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1/2·4-s + 1.13·7-s + 1.06·8-s − 0.603·11-s + 0.554·13-s − 0.801·14-s − 1/4·16-s − 0.970·17-s − 1.83·19-s + 0.426·22-s − 0.625·23-s − 0.392·26-s − 0.566·28-s − 0.185·29-s − 0.883·32-s + 0.685·34-s + 0.657·37-s + 1.29·38-s + 0.780·41-s + 1.21·43-s + 0.301·44-s + 0.442·46-s − 1.02·47-s + 2/7·49-s − 0.277·52-s + 0.274·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
good | 2 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−8.593385028447577936298463184160, −8.215283395565390706527454645557, −7.53137816266946138659022281649, −6.46027821051669619161653775760, −5.51078178070008390866658002727, −4.48379410243190882750827465918, −4.13767157383966277018539165853, −2.42827013674703527765723455521, −1.48649215064116892068636544934, 0,
1.48649215064116892068636544934, 2.42827013674703527765723455521, 4.13767157383966277018539165853, 4.48379410243190882750827465918, 5.51078178070008390866658002727, 6.46027821051669619161653775760, 7.53137816266946138659022281649, 8.215283395565390706527454645557, 8.593385028447577936298463184160