L(s) = 1 | − 4·11-s − 4·13-s + 2·17-s + 4·23-s + 8·31-s − 2·37-s + 2·41-s − 4·43-s + 4·47-s − 4·53-s − 4·59-s − 8·61-s + 4·67-s + 12·71-s + 4·73-s + 4·79-s − 8·83-s − 14·89-s + 12·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
L(s) = 1 | − 1.20·11-s − 1.10·13-s + 0.485·17-s + 0.834·23-s + 1.43·31-s − 0.328·37-s + 0.312·41-s − 0.609·43-s + 0.583·47-s − 0.549·53-s − 0.520·59-s − 1.02·61-s + 0.488·67-s + 1.42·71-s + 0.468·73-s + 0.450·79-s − 0.878·83-s − 1.48·89-s + 1.21·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 44100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 44100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−15.09680376096378, −14.27442710264256, −13.99689933975343, −13.35162267279013, −12.82772077387032, −12.35259959980058, −11.96357774333292, −11.21508640176267, −10.75163343546188, −10.18866095077759, −9.759017933654979, −9.248885851775646, −8.472823367554577, −7.988598482259317, −7.537923229402089, −6.942844465515559, −6.363427182005382, −5.584943343289440, −5.070826110215909, −4.689763615413025, −3.883225854737415, −2.926318894203262, −2.731002359539279, −1.852245234648200, −0.8905745780040668, 0,
0.8905745780040668, 1.852245234648200, 2.731002359539279, 2.926318894203262, 3.883225854737415, 4.689763615413025, 5.070826110215909, 5.584943343289440, 6.363427182005382, 6.942844465515559, 7.537923229402089, 7.988598482259317, 8.472823367554577, 9.248885851775646, 9.759017933654979, 10.18866095077759, 10.75163343546188, 11.21508640176267, 11.96357774333292, 12.35259959980058, 12.82772077387032, 13.35162267279013, 13.99689933975343, 14.27442710264256, 15.09680376096378