| L(s) = 1 | − 2·3-s + 5-s + 2·7-s + 9-s + 11-s − 6·13-s − 2·15-s − 17-s − 4·19-s − 4·21-s + 2·23-s + 25-s + 4·27-s + 2·29-s − 2·33-s + 2·35-s − 2·37-s + 12·39-s + 6·41-s − 10·43-s + 45-s − 6·47-s − 3·49-s + 2·51-s + 6·53-s + 55-s + 8·57-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.447·5-s + 0.755·7-s + 1/3·9-s + 0.301·11-s − 1.66·13-s − 0.516·15-s − 0.242·17-s − 0.917·19-s − 0.872·21-s + 0.417·23-s + 1/5·25-s + 0.769·27-s + 0.371·29-s − 0.348·33-s + 0.338·35-s − 0.328·37-s + 1.92·39-s + 0.937·41-s − 1.52·43-s + 0.149·45-s − 0.875·47-s − 3/7·49-s + 0.280·51-s + 0.824·53-s + 0.134·55-s + 1.05·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 59840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−14.61451165258798, −14.21982090183667, −13.49191472095579, −12.92323400201490, −12.44988269946064, −11.92359467715928, −11.63636750694643, −10.94748063823733, −10.66960323770226, −10.03155935329412, −9.578116490173095, −8.953331250940294, −8.304777720565496, −7.835900634534164, −7.042904686927533, −6.624336218994140, −6.217481077174583, −5.341933246336433, −5.073562393122910, −4.666639747527368, −3.959858143316773, −2.980697076909507, −2.293530590691728, −1.719180736100352, −0.7964289263311953, 0,
0.7964289263311953, 1.719180736100352, 2.293530590691728, 2.980697076909507, 3.959858143316773, 4.666639747527368, 5.073562393122910, 5.341933246336433, 6.217481077174583, 6.624336218994140, 7.042904686927533, 7.835900634534164, 8.304777720565496, 8.953331250940294, 9.578116490173095, 10.03155935329412, 10.66960323770226, 10.94748063823733, 11.63636750694643, 11.92359467715928, 12.44988269946064, 12.92323400201490, 13.49191472095579, 14.21982090183667, 14.61451165258798
