L(s) = 1 | − 3-s + 9-s + 3·11-s − 4·13-s − 6·17-s − 4·19-s + 3·23-s − 27-s + 3·29-s − 10·31-s − 3·33-s + 7·37-s + 4·39-s − 43-s + 12·47-s + 6·51-s − 6·53-s + 4·57-s + 12·59-s + 4·61-s − 7·67-s − 3·69-s − 9·71-s + 2·73-s − 17·79-s + 81-s + 6·83-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1/3·9-s + 0.904·11-s − 1.10·13-s − 1.45·17-s − 0.917·19-s + 0.625·23-s − 0.192·27-s + 0.557·29-s − 1.79·31-s − 0.522·33-s + 1.15·37-s + 0.640·39-s − 0.152·43-s + 1.75·47-s + 0.840·51-s − 0.824·53-s + 0.529·57-s + 1.56·59-s + 0.512·61-s − 0.855·67-s − 0.361·69-s − 1.06·71-s + 0.234·73-s − 1.91·79-s + 1/9·81-s + 0.658·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
good | 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 17 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 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
−14.55001833459328, −14.30270665615670, −13.38675742292024, −13.00326208764300, −12.62138609162010, −11.98813927527784, −11.55100091745012, −11.03822716945207, −10.63140269870989, −10.02521989401349, −9.388970580443072, −8.974299686502658, −8.542093622986796, −7.685685605308130, −7.028080115247691, −6.882077638265003, −6.116298229883079, −5.651961912184599, −4.904229976137339, −4.336206237366246, −4.049484938687878, −3.063904756161489, −2.310117171739363, −1.795316227418682, −0.8014456368984532, 0,
0.8014456368984532, 1.795316227418682, 2.310117171739363, 3.063904756161489, 4.049484938687878, 4.336206237366246, 4.904229976137339, 5.651961912184599, 6.116298229883079, 6.882077638265003, 7.028080115247691, 7.685685605308130, 8.542093622986796, 8.974299686502658, 9.388970580443072, 10.02521989401349, 10.63140269870989, 11.03822716945207, 11.55100091745012, 11.98813927527784, 12.62138609162010, 13.00326208764300, 13.38675742292024, 14.30270665615670, 14.55001833459328