| L(s) = 1 | + 3-s + 9-s + 2·11-s + 4·13-s + 5·17-s + 4·19-s + 5·23-s + 27-s − 6·29-s + 11·31-s + 2·33-s − 8·37-s + 4·39-s + 5·41-s − 47-s + 5·51-s − 12·53-s + 4·57-s + 2·59-s + 10·61-s + 5·69-s + 71-s − 2·73-s − 9·79-s + 81-s + 6·83-s − 6·87-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s + 0.603·11-s + 1.10·13-s + 1.21·17-s + 0.917·19-s + 1.04·23-s + 0.192·27-s − 1.11·29-s + 1.97·31-s + 0.348·33-s − 1.31·37-s + 0.640·39-s + 0.780·41-s − 0.145·47-s + 0.700·51-s − 1.64·53-s + 0.529·57-s + 0.260·59-s + 1.28·61-s + 0.601·69-s + 0.118·71-s − 0.234·73-s − 1.01·79-s + 1/9·81-s + 0.658·83-s − 0.643·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.660563396\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.660563396\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 11 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 9 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 11 T + p T^{2} \) |
| 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
−14.30444516896579, −13.98392668274541, −13.25774253867835, −13.01574886054581, −12.31129609030494, −11.73585618777038, −11.40745641299947, −10.73527910471914, −10.13625347959776, −9.724464296574877, −9.028514384616617, −8.824032230519404, −7.974535609263708, −7.757365946251080, −6.983497038238055, −6.510581572956485, −5.860352439486470, −5.285543493776674, −4.650032653945080, −3.907325999787016, −3.316651274432823, −3.049309442537253, −2.028275799132341, −1.273040530512444, −0.8142122824765387,
0.8142122824765387, 1.273040530512444, 2.028275799132341, 3.049309442537253, 3.316651274432823, 3.907325999787016, 4.650032653945080, 5.285543493776674, 5.860352439486470, 6.510581572956485, 6.983497038238055, 7.757365946251080, 7.974535609263708, 8.824032230519404, 9.028514384616617, 9.724464296574877, 10.13625347959776, 10.73527910471914, 11.40745641299947, 11.73585618777038, 12.31129609030494, 13.01574886054581, 13.25774253867835, 13.98392668274541, 14.30444516896579
