| L(s) = 1 | + 2·5-s − 7-s + 6·13-s + 2·17-s − 4·19-s + 2·23-s − 25-s − 8·29-s + 4·31-s − 2·35-s − 6·37-s − 10·41-s + 4·43-s − 4·47-s + 49-s − 4·53-s − 12·59-s + 2·61-s + 12·65-s + 12·67-s + 6·71-s + 2·73-s + 8·79-s + 4·85-s − 14·89-s − 6·91-s − 8·95-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 0.377·7-s + 1.66·13-s + 0.485·17-s − 0.917·19-s + 0.417·23-s − 1/5·25-s − 1.48·29-s + 0.718·31-s − 0.338·35-s − 0.986·37-s − 1.56·41-s + 0.609·43-s − 0.583·47-s + 1/7·49-s − 0.549·53-s − 1.56·59-s + 0.256·61-s + 1.48·65-s + 1.46·67-s + 0.712·71-s + 0.234·73-s + 0.900·79-s + 0.433·85-s − 1.48·89-s − 0.628·91-s − 0.820·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60984 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−14.38687803686912, −13.92614674557200, −13.58731785341869, −13.04068442602941, −12.71983626531487, −12.08960691506131, −11.36093154236952, −10.97815583674464, −10.49179516871130, −9.883377019094807, −9.512478117993125, −8.849209155542049, −8.478101789024329, −7.894859922815364, −7.156571512698862, −6.454076794830823, −6.238333075808200, −5.614410368918535, −5.115336976424993, −4.322564172845347, −3.514156386564112, −3.341259163136066, −2.258526358276191, −1.751347860486603, −1.076141521045050, 0,
1.076141521045050, 1.751347860486603, 2.258526358276191, 3.341259163136066, 3.514156386564112, 4.322564172845347, 5.115336976424993, 5.614410368918535, 6.238333075808200, 6.454076794830823, 7.156571512698862, 7.894859922815364, 8.478101789024329, 8.849209155542049, 9.512478117993125, 9.883377019094807, 10.49179516871130, 10.97815583674464, 11.36093154236952, 12.08960691506131, 12.71983626531487, 13.04068442602941, 13.58731785341869, 13.92614674557200, 14.38687803686912
