| L(s) = 1 | − 1.91·3-s − 3.56·5-s + 0.656·9-s + 11-s − 5.91·13-s + 6.82·15-s − 1.65·17-s − 1.48·19-s − 3.34·23-s + 7.73·25-s + 4.48·27-s + 3.08·29-s + 7.08·31-s − 1.91·33-s − 4.51·37-s + 11.3·39-s − 1.28·41-s − 1.59·43-s − 2.34·45-s + 1.65·47-s + 3.16·51-s + 9.22·53-s − 3.56·55-s + 2.83·57-s − 8.85·59-s − 6.68·61-s + 21.0·65-s + ⋯ |
| L(s) = 1 | − 1.10·3-s − 1.59·5-s + 0.218·9-s + 0.301·11-s − 1.63·13-s + 1.76·15-s − 0.401·17-s − 0.339·19-s − 0.697·23-s + 1.54·25-s + 0.862·27-s + 0.571·29-s + 1.27·31-s − 0.332·33-s − 0.741·37-s + 1.81·39-s − 0.200·41-s − 0.243·43-s − 0.349·45-s + 0.241·47-s + 0.443·51-s + 1.26·53-s − 0.481·55-s + 0.375·57-s − 1.15·59-s − 0.856·61-s + 2.61·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 + 1.91T + 3T^{2} \) |
| 5 | \( 1 + 3.56T + 5T^{2} \) |
| 13 | \( 1 + 5.91T + 13T^{2} \) |
| 17 | \( 1 + 1.65T + 17T^{2} \) |
| 19 | \( 1 + 1.48T + 19T^{2} \) |
| 23 | \( 1 + 3.34T + 23T^{2} \) |
| 29 | \( 1 - 3.08T + 29T^{2} \) |
| 31 | \( 1 - 7.08T + 31T^{2} \) |
| 37 | \( 1 + 4.51T + 37T^{2} \) |
| 41 | \( 1 + 1.28T + 41T^{2} \) |
| 43 | \( 1 + 1.59T + 43T^{2} \) |
| 47 | \( 1 - 1.65T + 47T^{2} \) |
| 53 | \( 1 - 9.22T + 53T^{2} \) |
| 59 | \( 1 + 8.85T + 59T^{2} \) |
| 61 | \( 1 + 6.68T + 61T^{2} \) |
| 67 | \( 1 - 9.82T + 67T^{2} \) |
| 71 | \( 1 - 8.61T + 71T^{2} \) |
| 73 | \( 1 - 4.56T + 73T^{2} \) |
| 79 | \( 1 - 6.39T + 79T^{2} \) |
| 83 | \( 1 + 0.167T + 83T^{2} \) |
| 89 | \( 1 + 2.56T + 89T^{2} \) |
| 97 | \( 1 - 9.73T + 97T^{2} \) |
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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
−7.32255429811050133652399069427, −6.78094595729083193335853895353, −6.16197851724481816273386821610, −5.14617601375481145809073928517, −4.69156912134922206940970029770, −4.09223180995199490008784113027, −3.17549620372375255231116719338, −2.25128378077648748407959038737, −0.74792191692199101347836075508, 0,
0.74792191692199101347836075508, 2.25128378077648748407959038737, 3.17549620372375255231116719338, 4.09223180995199490008784113027, 4.69156912134922206940970029770, 5.14617601375481145809073928517, 6.16197851724481816273386821610, 6.78094595729083193335853895353, 7.32255429811050133652399069427
