| L(s) = 1 | − 3-s − 2·4-s − 3.94·5-s + 9-s − 4.27·11-s + 2·12-s − 4.60·13-s + 3.94·15-s + 4·16-s − 2·17-s − 6.94·19-s + 7.88·20-s + 5.94·23-s + 10.5·25-s − 27-s + 7.88·29-s + 9.22·31-s + 4.27·33-s − 2·36-s − 37-s + 4.60·39-s − 41-s − 6.66·43-s + 8.55·44-s − 3.94·45-s − 2.93·47-s − 4·48-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 4-s − 1.76·5-s + 0.333·9-s − 1.29·11-s + 0.577·12-s − 1.27·13-s + 1.01·15-s + 16-s − 0.485·17-s − 1.59·19-s + 1.76·20-s + 1.23·23-s + 2.11·25-s − 0.192·27-s + 1.46·29-s + 1.65·31-s + 0.745·33-s − 0.333·36-s − 0.164·37-s + 0.737·39-s − 0.156·41-s − 1.01·43-s + 1.29·44-s − 0.587·45-s − 0.427·47-s − 0.577·48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 2 | \( 1 + 2T^{2} \) |
| 5 | \( 1 + 3.94T + 5T^{2} \) |
| 11 | \( 1 + 4.27T + 11T^{2} \) |
| 13 | \( 1 + 4.60T + 13T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 + 6.94T + 19T^{2} \) |
| 23 | \( 1 - 5.94T + 23T^{2} \) |
| 29 | \( 1 - 7.88T + 29T^{2} \) |
| 31 | \( 1 - 9.22T + 31T^{2} \) |
| 37 | \( 1 + T + 37T^{2} \) |
| 43 | \( 1 + 6.66T + 43T^{2} \) |
| 47 | \( 1 + 2.93T + 47T^{2} \) |
| 53 | \( 1 + 9.21T + 53T^{2} \) |
| 59 | \( 1 - 13.2T + 59T^{2} \) |
| 61 | \( 1 - 8.87T + 61T^{2} \) |
| 67 | \( 1 + 2.60T + 67T^{2} \) |
| 71 | \( 1 - 7.21T + 71T^{2} \) |
| 73 | \( 1 + 6.66T + 73T^{2} \) |
| 79 | \( 1 - 2.38T + 79T^{2} \) |
| 83 | \( 1 + 4.59T + 83T^{2} \) |
| 89 | \( 1 - 6.93T + 89T^{2} \) |
| 97 | \( 1 + 9.77T + 97T^{2} \) |
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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
−7.88429911819697170673413231270, −7.06550855759204635431204580473, −6.46529991545462644308780182470, −5.15442120296888724719273285726, −4.72848237468280045820264782656, −4.39082106293825466993555342403, −3.34680406097993601894936783108, −2.53164621767105562021420052121, −0.70958411150208033975693052155, 0,
0.70958411150208033975693052155, 2.53164621767105562021420052121, 3.34680406097993601894936783108, 4.39082106293825466993555342403, 4.72848237468280045820264782656, 5.15442120296888724719273285726, 6.46529991545462644308780182470, 7.06550855759204635431204580473, 7.88429911819697170673413231270
