L(s) = 1 | − 3-s + 3·5-s + 9-s + 11-s − 4·13-s − 3·15-s + 4·17-s − 8·23-s + 4·25-s − 27-s − 7·29-s − 11·31-s − 33-s + 4·37-s + 4·39-s − 4·41-s + 2·43-s + 3·45-s + 2·47-s − 4·51-s − 11·53-s + 3·55-s − 7·59-s + 10·61-s − 12·65-s − 10·67-s + 8·69-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1.34·5-s + 1/3·9-s + 0.301·11-s − 1.10·13-s − 0.774·15-s + 0.970·17-s − 1.66·23-s + 4/5·25-s − 0.192·27-s − 1.29·29-s − 1.97·31-s − 0.174·33-s + 0.657·37-s + 0.640·39-s − 0.624·41-s + 0.304·43-s + 0.447·45-s + 0.291·47-s − 0.560·51-s − 1.51·53-s + 0.404·55-s − 0.911·59-s + 1.28·61-s − 1.48·65-s − 1.22·67-s + 0.963·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 7 T + p T^{2} \) |
| 31 | \( 1 + 11 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 11 T + p T^{2} \) |
| 59 | \( 1 + 7 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 + 11 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−7.65139387270775445680771823733, −7.31061968140798128835743766750, −6.19075880460852636478648103084, −5.80837403510517046506211418897, −5.22275865910595504226501119198, −4.30653172754875403305622767693, −3.31739693425848685455371189910, −2.13248251863114793393283972617, −1.57617259056963537348571428711, 0,
1.57617259056963537348571428711, 2.13248251863114793393283972617, 3.31739693425848685455371189910, 4.30653172754875403305622767693, 5.22275865910595504226501119198, 5.80837403510517046506211418897, 6.19075880460852636478648103084, 7.31061968140798128835743766750, 7.65139387270775445680771823733