| L(s) = 1 | + 3·7-s − 3·11-s − 13-s + 7·17-s + 2·19-s − 23-s + 6·29-s − 10·31-s − 37-s + 41-s − 4·43-s + 4·47-s + 2·49-s + 53-s − 13·61-s − 8·67-s − 5·71-s + 10·73-s − 9·77-s − 79-s − 6·83-s − 9·89-s − 3·91-s − 97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | + 1.13·7-s − 0.904·11-s − 0.277·13-s + 1.69·17-s + 0.458·19-s − 0.208·23-s + 1.11·29-s − 1.79·31-s − 0.164·37-s + 0.156·41-s − 0.609·43-s + 0.583·47-s + 2/7·49-s + 0.137·53-s − 1.66·61-s − 0.977·67-s − 0.593·71-s + 1.17·73-s − 1.02·77-s − 0.112·79-s − 0.658·83-s − 0.953·89-s − 0.314·91-s − 0.101·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 187200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.492225323\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.492225323\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 - T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−13.10122404231050, −12.55310894506235, −12.15025296046000, −11.83108310457830, −11.18824080306874, −10.75188440371096, −10.38592938598105, −9.833897015854209, −9.431264538593155, −8.710218130318024, −8.301705299662606, −7.813974775154949, −7.425416816890052, −7.122765666518491, −6.177265564074338, −5.698138235311752, −5.262765351465811, −4.866410537254390, −4.316268563008902, −3.557824357823180, −3.079081271157544, −2.490937092515829, −1.706726167536048, −1.316436123491511, −0.4540172291342054,
0.4540172291342054, 1.316436123491511, 1.706726167536048, 2.490937092515829, 3.079081271157544, 3.557824357823180, 4.316268563008902, 4.866410537254390, 5.262765351465811, 5.698138235311752, 6.177265564074338, 7.122765666518491, 7.425416816890052, 7.813974775154949, 8.301705299662606, 8.710218130318024, 9.431264538593155, 9.833897015854209, 10.38592938598105, 10.75188440371096, 11.18824080306874, 11.83108310457830, 12.15025296046000, 12.55310894506235, 13.10122404231050
