L(s) = 1 | + 3-s + 5-s − 3·7-s + 9-s + 3·11-s + 13-s + 15-s − 3·17-s − 2·19-s − 3·21-s + 23-s + 25-s + 27-s − 8·29-s − 4·31-s + 3·33-s − 3·35-s − 5·37-s + 39-s − 7·41-s − 2·43-s + 45-s + 4·47-s + 2·49-s − 3·51-s − 11·53-s + 3·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.447·5-s − 1.13·7-s + 1/3·9-s + 0.904·11-s + 0.277·13-s + 0.258·15-s − 0.727·17-s − 0.458·19-s − 0.654·21-s + 0.208·23-s + 1/5·25-s + 0.192·27-s − 1.48·29-s − 0.718·31-s + 0.522·33-s − 0.507·35-s − 0.821·37-s + 0.160·39-s − 1.09·41-s − 0.304·43-s + 0.149·45-s + 0.583·47-s + 2/7·49-s − 0.420·51-s − 1.51·53-s + 0.404·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 - T \) |
good | 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 11 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 3 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 7 T + p T^{2} \) |
| 97 | \( 1 - 5 T + p T^{2} \) |
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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
−7.63043486985301838454125148375, −6.84831099539242363606853014688, −6.44782889387192106025642949344, −5.69463482156383282671794479483, −4.73383467927641961446495073583, −3.74719121504616644343009457381, −3.36991441308596354508568199767, −2.28697952013035090683506467255, −1.51611732496275347531337689403, 0,
1.51611732496275347531337689403, 2.28697952013035090683506467255, 3.36991441308596354508568199767, 3.74719121504616644343009457381, 4.73383467927641961446495073583, 5.69463482156383282671794479483, 6.44782889387192106025642949344, 6.84831099539242363606853014688, 7.63043486985301838454125148375