L(s) = 1 | − 2·2-s + 3-s + 2·4-s − 2·6-s − 3·7-s + 9-s − 11-s + 2·12-s + 13-s + 6·14-s − 4·16-s + 17-s − 2·18-s − 2·19-s − 3·21-s + 2·22-s + 3·23-s − 2·26-s + 27-s − 6·28-s − 2·29-s − 6·31-s + 8·32-s − 33-s − 2·34-s + 2·36-s − 11·37-s + ⋯ |
L(s) = 1 | − 1.41·2-s + 0.577·3-s + 4-s − 0.816·6-s − 1.13·7-s + 1/3·9-s − 0.301·11-s + 0.577·12-s + 0.277·13-s + 1.60·14-s − 16-s + 0.242·17-s − 0.471·18-s − 0.458·19-s − 0.654·21-s + 0.426·22-s + 0.625·23-s − 0.392·26-s + 0.192·27-s − 1.13·28-s − 0.371·29-s − 1.07·31-s + 1.41·32-s − 0.174·33-s − 0.342·34-s + 1/3·36-s − 1.80·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 2 | \( 1 + p T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 11 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 + 11 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 13 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 3 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 + 17 T + p T^{2} \) |
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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
−9.411799883495259178827358194573, −8.895634882857946943349810693110, −8.166887923701654716624614272709, −7.20815108696393773974523839509, −6.66509580772906665088755572501, −5.36708499482522794774379461836, −3.92430597429407407364736612147, −2.89539799534326025086395161933, −1.63747587545011337875447500553, 0,
1.63747587545011337875447500553, 2.89539799534326025086395161933, 3.92430597429407407364736612147, 5.36708499482522794774379461836, 6.66509580772906665088755572501, 7.20815108696393773974523839509, 8.166887923701654716624614272709, 8.895634882857946943349810693110, 9.411799883495259178827358194573