| L(s) = 1 | − 3-s + 5-s + 4·7-s + 9-s + 13-s − 15-s + 2·17-s + 8·19-s − 4·21-s + 4·23-s + 25-s − 27-s + 6·29-s − 8·31-s + 4·35-s + 6·37-s − 39-s − 6·41-s − 4·43-s + 45-s + 4·47-s + 9·49-s − 2·51-s + 6·53-s − 8·57-s + 8·59-s − 10·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1.51·7-s + 1/3·9-s + 0.277·13-s − 0.258·15-s + 0.485·17-s + 1.83·19-s − 0.872·21-s + 0.834·23-s + 1/5·25-s − 0.192·27-s + 1.11·29-s − 1.43·31-s + 0.676·35-s + 0.986·37-s − 0.160·39-s − 0.937·41-s − 0.609·43-s + 0.149·45-s + 0.583·47-s + 9/7·49-s − 0.280·51-s + 0.824·53-s − 1.05·57-s + 1.04·59-s − 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.623222105\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.623222105\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−7.944490996203046871006740236534, −7.37195814834809496724933212125, −6.68191318195853732161710038721, −5.63118185057842449000250085630, −5.28635645982431779207741724983, −4.68425346732578023214280853595, −3.69498404820957952992327762228, −2.70189473146539199878359027873, −1.56097181112985290639590099657, −0.991953726160196555799716449956,
0.991953726160196555799716449956, 1.56097181112985290639590099657, 2.70189473146539199878359027873, 3.69498404820957952992327762228, 4.68425346732578023214280853595, 5.28635645982431779207741724983, 5.63118185057842449000250085630, 6.68191318195853732161710038721, 7.37195814834809496724933212125, 7.944490996203046871006740236534
