| L(s) = 1 | + 3·7-s − 4·11-s + 7·13-s + 4·17-s − 19-s − 8·23-s − 3·31-s − 2·37-s + 6·41-s + 11·43-s + 6·47-s + 2·49-s + 6·53-s + 6·59-s − 61-s + 15·67-s − 6·71-s − 2·73-s − 12·77-s − 8·79-s + 2·83-s + 16·89-s + 21·91-s − 13·97-s + 2·101-s + 16·103-s − 18·107-s + ⋯ |
| L(s) = 1 | + 1.13·7-s − 1.20·11-s + 1.94·13-s + 0.970·17-s − 0.229·19-s − 1.66·23-s − 0.538·31-s − 0.328·37-s + 0.937·41-s + 1.67·43-s + 0.875·47-s + 2/7·49-s + 0.824·53-s + 0.781·59-s − 0.128·61-s + 1.83·67-s − 0.712·71-s − 0.234·73-s − 1.36·77-s − 0.900·79-s + 0.219·83-s + 1.69·89-s + 2.20·91-s − 1.31·97-s + 0.199·101-s + 1.57·103-s − 1.74·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.464976709\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.464976709\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 7 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 - 15 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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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
−7.82883755733275768109541723260, −7.59214193016505865303336188762, −6.38158274543910024618894947873, −5.68243108631662397030125068813, −5.31722279045713475378828135784, −4.18934854058334690225928623151, −3.75497666461901986247681871396, −2.61094601593440410215203063696, −1.77469956621102820450984743376, −0.828355383526682712238264133479,
0.828355383526682712238264133479, 1.77469956621102820450984743376, 2.61094601593440410215203063696, 3.75497666461901986247681871396, 4.18934854058334690225928623151, 5.31722279045713475378828135784, 5.68243108631662397030125068813, 6.38158274543910024618894947873, 7.59214193016505865303336188762, 7.82883755733275768109541723260
