| L(s) = 1 | − 3-s + 0.324·5-s − 4.96·7-s + 9-s + 2.15·11-s + 2·13-s − 0.324·15-s + 5.02·17-s + 1.61·19-s + 4.96·21-s − 8·23-s − 4.89·25-s − 27-s + 2.31·29-s − 1.61·31-s − 2.15·33-s − 1.61·35-s − 6.96·37-s − 2·39-s + 9.86·41-s − 4.06·43-s + 0.324·45-s + 10.4·47-s + 17.6·49-s − 5.02·51-s − 4.89·53-s + 0.700·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.145·5-s − 1.87·7-s + 0.333·9-s + 0.650·11-s + 0.554·13-s − 0.0838·15-s + 1.21·17-s + 0.369·19-s + 1.08·21-s − 1.66·23-s − 0.978·25-s − 0.192·27-s + 0.429·29-s − 0.289·31-s − 0.375·33-s − 0.272·35-s − 1.14·37-s − 0.320·39-s + 1.54·41-s − 0.619·43-s + 0.0484·45-s + 1.52·47-s + 2.51·49-s − 0.703·51-s − 0.672·53-s + 0.0944·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 167 | \( 1 - T \) |
| good | 5 | \( 1 - 0.324T + 5T^{2} \) |
| 7 | \( 1 + 4.96T + 7T^{2} \) |
| 11 | \( 1 - 2.15T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 - 5.02T + 17T^{2} \) |
| 19 | \( 1 - 1.61T + 19T^{2} \) |
| 23 | \( 1 + 8T + 23T^{2} \) |
| 29 | \( 1 - 2.31T + 29T^{2} \) |
| 31 | \( 1 + 1.61T + 31T^{2} \) |
| 37 | \( 1 + 6.96T + 37T^{2} \) |
| 41 | \( 1 - 9.86T + 41T^{2} \) |
| 43 | \( 1 + 4.06T + 43T^{2} \) |
| 47 | \( 1 - 10.4T + 47T^{2} \) |
| 53 | \( 1 + 4.89T + 53T^{2} \) |
| 59 | \( 1 - 8.31T + 59T^{2} \) |
| 61 | \( 1 + 0.0303T + 61T^{2} \) |
| 67 | \( 1 + 6.71T + 67T^{2} \) |
| 71 | \( 1 + 8.12T + 71T^{2} \) |
| 73 | \( 1 - 4.26T + 73T^{2} \) |
| 79 | \( 1 - 15.3T + 79T^{2} \) |
| 83 | \( 1 + 5.79T + 83T^{2} \) |
| 89 | \( 1 - 12.2T + 89T^{2} \) |
| 97 | \( 1 + 8.05T + 97T^{2} \) |
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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
−7.969333816856581382141590735595, −7.22817135156938466297701531055, −6.37729859858106436269035302059, −6.01957869263193211444301971971, −5.37336160214590669487915737762, −3.90605862444746094724185916555, −3.69112474325232499769050563751, −2.55154450983491828023189593789, −1.22290331148981767870813992561, 0,
1.22290331148981767870813992561, 2.55154450983491828023189593789, 3.69112474325232499769050563751, 3.90605862444746094724185916555, 5.37336160214590669487915737762, 6.01957869263193211444301971971, 6.37729859858106436269035302059, 7.22817135156938466297701531055, 7.969333816856581382141590735595
