| L(s) = 1 | − 2.56·3-s − 5-s + 4.56·7-s + 3.56·9-s − 4·11-s + 2.56·13-s + 2.56·15-s − 0.561·17-s − 19-s − 11.6·21-s − 5.68·23-s + 25-s − 1.43·27-s − 3.43·29-s − 5.12·31-s + 10.2·33-s − 4.56·35-s − 1.12·37-s − 6.56·39-s + 8.24·41-s − 2·43-s − 3.56·45-s + 3.12·47-s + 13.8·49-s + 1.43·51-s + 6.56·53-s + 4·55-s + ⋯ |
| L(s) = 1 | − 1.47·3-s − 0.447·5-s + 1.72·7-s + 1.18·9-s − 1.20·11-s + 0.710·13-s + 0.661·15-s − 0.136·17-s − 0.229·19-s − 2.54·21-s − 1.18·23-s + 0.200·25-s − 0.276·27-s − 0.638·29-s − 0.920·31-s + 1.78·33-s − 0.771·35-s − 0.184·37-s − 1.05·39-s + 1.28·41-s − 0.304·43-s − 0.530·45-s + 0.455·47-s + 1.97·49-s + 0.201·51-s + 0.901·53-s + 0.539·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + 2.56T + 3T^{2} \) |
| 7 | \( 1 - 4.56T + 7T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 - 2.56T + 13T^{2} \) |
| 17 | \( 1 + 0.561T + 17T^{2} \) |
| 23 | \( 1 + 5.68T + 23T^{2} \) |
| 29 | \( 1 + 3.43T + 29T^{2} \) |
| 31 | \( 1 + 5.12T + 31T^{2} \) |
| 37 | \( 1 + 1.12T + 37T^{2} \) |
| 41 | \( 1 - 8.24T + 41T^{2} \) |
| 43 | \( 1 + 2T + 43T^{2} \) |
| 47 | \( 1 - 3.12T + 47T^{2} \) |
| 53 | \( 1 - 6.56T + 53T^{2} \) |
| 59 | \( 1 - 12.8T + 59T^{2} \) |
| 61 | \( 1 + 4.87T + 61T^{2} \) |
| 67 | \( 1 + 10.5T + 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 - 14.8T + 73T^{2} \) |
| 79 | \( 1 - 5.12T + 79T^{2} \) |
| 83 | \( 1 + 14T + 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 + 1.12T + 97T^{2} \) |
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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
−8.127076946012730229522224917624, −7.66584508476188592531616769520, −6.82788807543498151207696316201, −5.60630922824148966745539700575, −5.54608592635643997986852820133, −4.55872822282472064507499764032, −3.95589789740851047649235296684, −2.34176164250424344749407496506, −1.27561062388972110450253528605, 0,
1.27561062388972110450253528605, 2.34176164250424344749407496506, 3.95589789740851047649235296684, 4.55872822282472064507499764032, 5.54608592635643997986852820133, 5.60630922824148966745539700575, 6.82788807543498151207696316201, 7.66584508476188592531616769520, 8.127076946012730229522224917624
