L(s) = 1 | + 5-s − 2·7-s − 4·11-s + 2·17-s + 19-s + 2·23-s + 25-s + 6·29-s − 2·35-s − 8·37-s + 2·41-s − 6·43-s + 2·47-s − 3·49-s + 4·53-s − 4·55-s − 4·59-s − 10·61-s − 12·67-s − 8·71-s − 2·73-s + 8·77-s − 8·79-s − 14·83-s + 2·85-s + 2·89-s + 95-s + ⋯ |
L(s) = 1 | + 0.447·5-s − 0.755·7-s − 1.20·11-s + 0.485·17-s + 0.229·19-s + 0.417·23-s + 1/5·25-s + 1.11·29-s − 0.338·35-s − 1.31·37-s + 0.312·41-s − 0.914·43-s + 0.291·47-s − 3/7·49-s + 0.549·53-s − 0.539·55-s − 0.520·59-s − 1.28·61-s − 1.46·67-s − 0.949·71-s − 0.234·73-s + 0.911·77-s − 0.900·79-s − 1.53·83-s + 0.216·85-s + 0.211·89-s + 0.102·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 19 | \( 1 - T \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−8.267336619025173410548977907843, −7.43682513460105661275697768763, −6.75659824240593650566848376652, −5.91430701199690925597911387022, −5.28370670565256217450503955696, −4.46110001219319117486236036108, −3.21102826685750890761826357065, −2.74839164763387172087967551112, −1.47064461946244187377631113071, 0,
1.47064461946244187377631113071, 2.74839164763387172087967551112, 3.21102826685750890761826357065, 4.46110001219319117486236036108, 5.28370670565256217450503955696, 5.91430701199690925597911387022, 6.75659824240593650566848376652, 7.43682513460105661275697768763, 8.267336619025173410548977907843