| L(s) = 1 | − 1.94·2-s − 3-s + 1.77·4-s − 2.73·5-s + 1.94·6-s + 0.439·8-s + 9-s + 5.31·10-s − 4.46·11-s − 1.77·12-s − 6.87·13-s + 2.73·15-s − 4.40·16-s − 4.96·17-s − 1.94·18-s + 1.89·19-s − 4.85·20-s + 8.68·22-s + 8.36·23-s − 0.439·24-s + 2.49·25-s + 13.3·26-s − 27-s + 5.40·29-s − 5.31·30-s − 9.94·31-s + 7.67·32-s + ⋯ |
| L(s) = 1 | − 1.37·2-s − 0.577·3-s + 0.886·4-s − 1.22·5-s + 0.793·6-s + 0.155·8-s + 0.333·9-s + 1.68·10-s − 1.34·11-s − 0.512·12-s − 1.90·13-s + 0.707·15-s − 1.10·16-s − 1.20·17-s − 0.457·18-s + 0.435·19-s − 1.08·20-s + 1.85·22-s + 1.74·23-s − 0.0897·24-s + 0.499·25-s + 2.61·26-s − 0.192·27-s + 1.00·29-s − 0.971·30-s − 1.78·31-s + 1.35·32-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 | \( 1 + 1.94T + 2T^{2} \) |
| 5 | \( 1 + 2.73T + 5T^{2} \) |
| 11 | \( 1 + 4.46T + 11T^{2} \) |
| 13 | \( 1 + 6.87T + 13T^{2} \) |
| 17 | \( 1 + 4.96T + 17T^{2} \) |
| 19 | \( 1 - 1.89T + 19T^{2} \) |
| 23 | \( 1 - 8.36T + 23T^{2} \) |
| 29 | \( 1 - 5.40T + 29T^{2} \) |
| 31 | \( 1 + 9.94T + 31T^{2} \) |
| 37 | \( 1 - 3.92T + 37T^{2} \) |
| 43 | \( 1 + 0.181T + 43T^{2} \) |
| 47 | \( 1 - 12.6T + 47T^{2} \) |
| 53 | \( 1 - 0.454T + 53T^{2} \) |
| 59 | \( 1 - 3.12T + 59T^{2} \) |
| 61 | \( 1 + 1.59T + 61T^{2} \) |
| 67 | \( 1 + 8.14T + 67T^{2} \) |
| 71 | \( 1 - 9.16T + 71T^{2} \) |
| 73 | \( 1 - 7.69T + 73T^{2} \) |
| 79 | \( 1 + 8.24T + 79T^{2} \) |
| 83 | \( 1 + 5.84T + 83T^{2} \) |
| 89 | \( 1 + 9.20T + 89T^{2} \) |
| 97 | \( 1 - 14.3T + 97T^{2} \) |
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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
−7.61221334455924131481495169342, −7.30346476647197260322366355209, −6.85793506448067251686697387344, −5.46558869117265580624222156675, −4.82216941807531668069864528868, −4.26865408420077084112645782602, −2.91304679799412932981088269259, −2.18656194267812491373138469285, −0.70603827444624321998484545418, 0,
0.70603827444624321998484545418, 2.18656194267812491373138469285, 2.91304679799412932981088269259, 4.26865408420077084112645782602, 4.82216941807531668069864528868, 5.46558869117265580624222156675, 6.85793506448067251686697387344, 7.30346476647197260322366355209, 7.61221334455924131481495169342
