| L(s) = 1 | − 2·5-s + 11-s − 2·13-s − 2·17-s + 4·19-s + 8·23-s − 25-s + 2·29-s + 4·31-s + 6·37-s + 6·41-s − 4·43-s − 12·47-s + 10·53-s − 2·55-s + 8·59-s − 10·61-s + 4·65-s + 4·67-s + 8·71-s + 2·73-s − 8·79-s + 12·83-s + 4·85-s + 10·89-s − 8·95-s − 10·97-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.301·11-s − 0.554·13-s − 0.485·17-s + 0.917·19-s + 1.66·23-s − 1/5·25-s + 0.371·29-s + 0.718·31-s + 0.986·37-s + 0.937·41-s − 0.609·43-s − 1.75·47-s + 1.37·53-s − 0.269·55-s + 1.04·59-s − 1.28·61-s + 0.496·65-s + 0.488·67-s + 0.949·71-s + 0.234·73-s − 0.900·79-s + 1.31·83-s + 0.433·85-s + 1.05·89-s − 0.820·95-s − 1.01·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38808 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38808 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.845538392\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.845538392\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 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 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−14.91173970740452, −14.42293281658289, −13.74616240306755, −13.22140886982817, −12.77715435083191, −12.03029357880463, −11.75576066213999, −11.19290896716912, −10.80964820237757, −9.908530596775590, −9.610240113204548, −8.920520802760511, −8.388662013990599, −7.771198976069836, −7.340085079167872, −6.742324144164662, −6.230267137074299, −5.307423186502689, −4.857005021401335, −4.262739679474527, −3.580010506780156, −2.965280475442103, −2.320671204385505, −1.236503897037112, −0.5524918062657641,
0.5524918062657641, 1.236503897037112, 2.320671204385505, 2.965280475442103, 3.580010506780156, 4.262739679474527, 4.857005021401335, 5.307423186502689, 6.230267137074299, 6.742324144164662, 7.340085079167872, 7.771198976069836, 8.388662013990599, 8.920520802760511, 9.610240113204548, 9.908530596775590, 10.80964820237757, 11.19290896716912, 11.75576066213999, 12.03029357880463, 12.77715435083191, 13.22140886982817, 13.74616240306755, 14.42293281658289, 14.91173970740452
