| L(s) = 1 | − 7-s + 2·13-s − 2·19-s − 6·29-s + 8·31-s − 4·37-s + 6·41-s + 2·43-s + 6·47-s + 49-s + 6·53-s − 12·59-s − 8·61-s + 2·67-s + 6·71-s − 2·73-s − 16·79-s + 6·89-s − 2·91-s + 10·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 0.377·7-s + 0.554·13-s − 0.458·19-s − 1.11·29-s + 1.43·31-s − 0.657·37-s + 0.937·41-s + 0.304·43-s + 0.875·47-s + 1/7·49-s + 0.824·53-s − 1.56·59-s − 1.02·61-s + 0.244·67-s + 0.712·71-s − 0.234·73-s − 1.80·79-s + 0.635·89-s − 0.209·91-s + 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−13.91927122191013, −13.49032147436007, −13.08083446700939, −12.49758472418832, −12.08821667709205, −11.60740942566402, −10.88078386126314, −10.68074021065073, −10.10366339313679, −9.401644204170805, −9.173968038505438, −8.485970917605211, −8.076419625244408, −7.416954819360550, −6.981720300022348, −6.317536567885902, −5.909609001642564, −5.432981982431845, −4.593046073301202, −4.196775047498180, −3.565868851555911, −2.943888788179203, −2.354459311251544, −1.597569159008783, −0.8700925357743639, 0,
0.8700925357743639, 1.597569159008783, 2.354459311251544, 2.943888788179203, 3.565868851555911, 4.196775047498180, 4.593046073301202, 5.432981982431845, 5.909609001642564, 6.317536567885902, 6.981720300022348, 7.416954819360550, 8.076419625244408, 8.485970917605211, 9.173968038505438, 9.401644204170805, 10.10366339313679, 10.68074021065073, 10.88078386126314, 11.60740942566402, 12.08821667709205, 12.49758472418832, 13.08083446700939, 13.49032147436007, 13.91927122191013
