| L(s) = 1 | + 7-s − 6·11-s − 5·13-s − 6·17-s + 5·19-s − 6·23-s + 6·29-s − 31-s − 2·37-s + 43-s + 6·47-s − 6·49-s − 12·53-s + 6·59-s − 13·61-s − 11·67-s − 2·73-s − 6·77-s + 8·79-s − 6·83-s − 5·91-s + 7·97-s + 12·101-s + 4·103-s + 12·107-s − 7·109-s + 12·113-s + ⋯ |
| L(s) = 1 | + 0.377·7-s − 1.80·11-s − 1.38·13-s − 1.45·17-s + 1.14·19-s − 1.25·23-s + 1.11·29-s − 0.179·31-s − 0.328·37-s + 0.152·43-s + 0.875·47-s − 6/7·49-s − 1.64·53-s + 0.781·59-s − 1.66·61-s − 1.34·67-s − 0.234·73-s − 0.683·77-s + 0.900·79-s − 0.658·83-s − 0.524·91-s + 0.710·97-s + 1.19·101-s + 0.394·103-s + 1.16·107-s − 0.670·109-s + 1.12·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 67 | \( 1 + 11 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−9.853259570123841667190565291355, −8.821114032953874975275414885204, −7.82946935002197249360095495788, −7.40049250827422786308554046568, −6.17945227400387495932931920930, −5.09151237655519270484018689309, −4.57579314482224407000052258764, −2.97401840613225752286969945588, −2.10805618754928013340727410523, 0,
2.10805618754928013340727410523, 2.97401840613225752286969945588, 4.57579314482224407000052258764, 5.09151237655519270484018689309, 6.17945227400387495932931920930, 7.40049250827422786308554046568, 7.82946935002197249360095495788, 8.821114032953874975275414885204, 9.853259570123841667190565291355
