| L(s) = 1 | − 4·7-s − 4·13-s − 2·17-s − 4·19-s − 8·23-s − 5·25-s − 8·29-s + 4·31-s − 4·37-s + 6·41-s + 4·43-s − 8·47-s + 9·49-s + 8·53-s − 12·59-s − 12·61-s − 12·67-s + 8·71-s + 6·73-s − 4·79-s + 4·83-s + 6·89-s + 16·91-s − 2·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 1.51·7-s − 1.10·13-s − 0.485·17-s − 0.917·19-s − 1.66·23-s − 25-s − 1.48·29-s + 0.718·31-s − 0.657·37-s + 0.937·41-s + 0.609·43-s − 1.16·47-s + 9/7·49-s + 1.09·53-s − 1.56·59-s − 1.53·61-s − 1.46·67-s + 0.949·71-s + 0.702·73-s − 0.450·79-s + 0.439·83-s + 0.635·89-s + 1.67·91-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 278784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 278784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−12.86924800742994, −12.54430246257724, −12.13105770249125, −11.78125952342882, −11.11139037631177, −10.52153187383629, −10.26098512882763, −9.624875898205067, −9.421974838970338, −9.026760076330328, −8.268960221022524, −7.781314724586858, −7.429693453138201, −6.768873872003985, −6.408091233417096, −5.860160310470394, −5.643979718839939, −4.694543844655569, −4.307965265846679, −3.785467679966550, −3.275940759142665, −2.616156986581181, −2.147997016181483, −1.635163777428287, −0.4045019431506272, 0,
0.4045019431506272, 1.635163777428287, 2.147997016181483, 2.616156986581181, 3.275940759142665, 3.785467679966550, 4.307965265846679, 4.694543844655569, 5.643979718839939, 5.860160310470394, 6.408091233417096, 6.768873872003985, 7.429693453138201, 7.781314724586858, 8.268960221022524, 9.026760076330328, 9.421974838970338, 9.624875898205067, 10.26098512882763, 10.52153187383629, 11.11139037631177, 11.78125952342882, 12.13105770249125, 12.54430246257724, 12.86924800742994
