| L(s) = 1 | + 2-s + 2.28·3-s + 4-s − 5-s + 2.28·6-s − 1.26·7-s + 8-s + 2.22·9-s − 10-s + 11-s + 2.28·12-s − 2.76·13-s − 1.26·14-s − 2.28·15-s + 16-s − 5.74·17-s + 2.22·18-s − 6.36·19-s − 20-s − 2.89·21-s + 22-s − 8.38·23-s + 2.28·24-s + 25-s − 2.76·26-s − 1.76·27-s − 1.26·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.32·3-s + 0.5·4-s − 0.447·5-s + 0.933·6-s − 0.479·7-s + 0.353·8-s + 0.743·9-s − 0.316·10-s + 0.301·11-s + 0.660·12-s − 0.766·13-s − 0.338·14-s − 0.590·15-s + 0.250·16-s − 1.39·17-s + 0.525·18-s − 1.45·19-s − 0.223·20-s − 0.632·21-s + 0.213·22-s − 1.74·23-s + 0.466·24-s + 0.200·25-s − 0.541·26-s − 0.339·27-s − 0.239·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 43 | \( 1 + T \) |
| good | 3 | \( 1 - 2.28T + 3T^{2} \) |
| 7 | \( 1 + 1.26T + 7T^{2} \) |
| 13 | \( 1 + 2.76T + 13T^{2} \) |
| 17 | \( 1 + 5.74T + 17T^{2} \) |
| 19 | \( 1 + 6.36T + 19T^{2} \) |
| 23 | \( 1 + 8.38T + 23T^{2} \) |
| 29 | \( 1 + 2.38T + 29T^{2} \) |
| 31 | \( 1 + 7.45T + 31T^{2} \) |
| 37 | \( 1 - 1.43T + 37T^{2} \) |
| 41 | \( 1 - 10.7T + 41T^{2} \) |
| 47 | \( 1 - 9.64T + 47T^{2} \) |
| 53 | \( 1 + 13.2T + 53T^{2} \) |
| 59 | \( 1 - 6.07T + 59T^{2} \) |
| 61 | \( 1 - 2.31T + 61T^{2} \) |
| 67 | \( 1 - 3.53T + 67T^{2} \) |
| 71 | \( 1 - 4.44T + 71T^{2} \) |
| 73 | \( 1 - 12.7T + 73T^{2} \) |
| 79 | \( 1 - 3.89T + 79T^{2} \) |
| 83 | \( 1 - 2.71T + 83T^{2} \) |
| 89 | \( 1 + 8.51T + 89T^{2} \) |
| 97 | \( 1 - 9.97T + 97T^{2} \) |
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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
−7.931356373202275202510133490063, −7.27347292072587258192427467646, −6.51255741450964669757223486065, −5.82952320762740132634831019056, −4.60508731024014062517634711228, −4.02623614741504547693707377190, −3.49058855121436885058513822214, −2.32257412692589964161560573629, −2.11725151192317186352715248565, 0,
2.11725151192317186352715248565, 2.32257412692589964161560573629, 3.49058855121436885058513822214, 4.02623614741504547693707377190, 4.60508731024014062517634711228, 5.82952320762740132634831019056, 6.51255741450964669757223486065, 7.27347292072587258192427467646, 7.931356373202275202510133490063
