| L(s) = 1 | − 2.93·5-s − 2.93·7-s + 1.68·11-s − 5.57·13-s − 0.319·17-s − 4.93·19-s − 0.939·23-s + 3.63·25-s + 2.31·29-s − 7.89·31-s + 8.63·35-s + 5.89·37-s + 41-s + 9.77·43-s + 4.31·47-s + 1.63·49-s + 4·53-s − 4.93·55-s + 2.12·59-s − 9.89·61-s + 16.3·65-s − 4.63·67-s − 2.19·71-s − 5.89·73-s − 4.93·77-s + 3.27·79-s + 2.42·83-s + ⋯ |
| L(s) = 1 | − 1.31·5-s − 1.11·7-s + 0.506·11-s − 1.54·13-s − 0.0775·17-s − 1.13·19-s − 0.195·23-s + 0.727·25-s + 0.430·29-s − 1.41·31-s + 1.46·35-s + 0.969·37-s + 0.156·41-s + 1.49·43-s + 0.630·47-s + 0.234·49-s + 0.549·53-s − 0.666·55-s + 0.276·59-s − 1.26·61-s + 2.03·65-s − 0.566·67-s − 0.260·71-s − 0.690·73-s − 0.562·77-s + 0.368·79-s + 0.265·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2952 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2952 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5996757578\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5996757578\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 5 | \( 1 + 2.93T + 5T^{2} \) |
| 7 | \( 1 + 2.93T + 7T^{2} \) |
| 11 | \( 1 - 1.68T + 11T^{2} \) |
| 13 | \( 1 + 5.57T + 13T^{2} \) |
| 17 | \( 1 + 0.319T + 17T^{2} \) |
| 19 | \( 1 + 4.93T + 19T^{2} \) |
| 23 | \( 1 + 0.939T + 23T^{2} \) |
| 29 | \( 1 - 2.31T + 29T^{2} \) |
| 31 | \( 1 + 7.89T + 31T^{2} \) |
| 37 | \( 1 - 5.89T + 37T^{2} \) |
| 43 | \( 1 - 9.77T + 43T^{2} \) |
| 47 | \( 1 - 4.31T + 47T^{2} \) |
| 53 | \( 1 - 4T + 53T^{2} \) |
| 59 | \( 1 - 2.12T + 59T^{2} \) |
| 61 | \( 1 + 9.89T + 61T^{2} \) |
| 67 | \( 1 + 4.63T + 67T^{2} \) |
| 71 | \( 1 + 2.19T + 71T^{2} \) |
| 73 | \( 1 + 5.89T + 73T^{2} \) |
| 79 | \( 1 - 3.27T + 79T^{2} \) |
| 83 | \( 1 - 2.42T + 83T^{2} \) |
| 89 | \( 1 - 9.75T + 89T^{2} \) |
| 97 | \( 1 - 6.30T + 97T^{2} \) |
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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
−8.867677466839274772004161786856, −7.77528133899920389055910315082, −7.36511366601676682325660602799, −6.59436869822157013438168147598, −5.79827141421872658827567079349, −4.58623594823978499514361720300, −4.06020751411606703077124932338, −3.19001661191908471091190597993, −2.25609920301484843796124378548, −0.44565688588095577215572947939,
0.44565688588095577215572947939, 2.25609920301484843796124378548, 3.19001661191908471091190597993, 4.06020751411606703077124932338, 4.58623594823978499514361720300, 5.79827141421872658827567079349, 6.59436869822157013438168147598, 7.36511366601676682325660602799, 7.77528133899920389055910315082, 8.867677466839274772004161786856
