| L(s) = 1 | + 0.463·3-s + 1.98·5-s − 7-s − 2.78·9-s + 0.628·11-s + 3.19·13-s + 0.919·15-s − 4.85·17-s + 0.161·19-s − 0.463·21-s − 3.20·23-s − 1.06·25-s − 2.68·27-s + 1.41·29-s + 5.34·31-s + 0.291·33-s − 1.98·35-s + 2.87·37-s + 1.47·39-s − 9.57·41-s + 9.71·43-s − 5.52·45-s − 9.70·47-s + 49-s − 2.25·51-s − 10.8·53-s + 1.24·55-s + ⋯ |
| L(s) = 1 | + 0.267·3-s + 0.886·5-s − 0.377·7-s − 0.928·9-s + 0.189·11-s + 0.884·13-s + 0.237·15-s − 1.17·17-s + 0.0369·19-s − 0.101·21-s − 0.668·23-s − 0.213·25-s − 0.516·27-s + 0.261·29-s + 0.959·31-s + 0.0507·33-s − 0.335·35-s + 0.473·37-s + 0.236·39-s − 1.49·41-s + 1.48·43-s − 0.823·45-s − 1.41·47-s + 0.142·49-s − 0.315·51-s − 1.48·53-s + 0.167·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 3 | \( 1 - 0.463T + 3T^{2} \) |
| 5 | \( 1 - 1.98T + 5T^{2} \) |
| 11 | \( 1 - 0.628T + 11T^{2} \) |
| 13 | \( 1 - 3.19T + 13T^{2} \) |
| 17 | \( 1 + 4.85T + 17T^{2} \) |
| 19 | \( 1 - 0.161T + 19T^{2} \) |
| 23 | \( 1 + 3.20T + 23T^{2} \) |
| 29 | \( 1 - 1.41T + 29T^{2} \) |
| 31 | \( 1 - 5.34T + 31T^{2} \) |
| 37 | \( 1 - 2.87T + 37T^{2} \) |
| 41 | \( 1 + 9.57T + 41T^{2} \) |
| 43 | \( 1 - 9.71T + 43T^{2} \) |
| 47 | \( 1 + 9.70T + 47T^{2} \) |
| 53 | \( 1 + 10.8T + 53T^{2} \) |
| 59 | \( 1 + 2.12T + 59T^{2} \) |
| 61 | \( 1 - 2.46T + 61T^{2} \) |
| 67 | \( 1 - 12.6T + 67T^{2} \) |
| 71 | \( 1 + 7.18T + 71T^{2} \) |
| 73 | \( 1 + 9.04T + 73T^{2} \) |
| 79 | \( 1 + 9.58T + 79T^{2} \) |
| 83 | \( 1 + 7.49T + 83T^{2} \) |
| 89 | \( 1 + 2.49T + 89T^{2} \) |
| 97 | \( 1 - 5.89T + 97T^{2} \) |
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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
−7.67371694241126296045960679234, −6.58528625631522080926547613804, −6.24675936611761190463019075561, −5.66059622414943035798434749276, −4.73961823471125511702843122479, −3.89133182222199756029774871465, −3.03016531571004761139411292819, −2.32369624269412193386358783473, −1.44072611582104409547701424425, 0,
1.44072611582104409547701424425, 2.32369624269412193386358783473, 3.03016531571004761139411292819, 3.89133182222199756029774871465, 4.73961823471125511702843122479, 5.66059622414943035798434749276, 6.24675936611761190463019075561, 6.58528625631522080926547613804, 7.67371694241126296045960679234
