| L(s) = 1 | − 2-s − 4-s − 5-s + 4·7-s + 3·8-s + 10-s − 4·14-s − 16-s + 4·17-s + 4·19-s + 20-s + 8·23-s + 25-s − 4·28-s + 8·29-s − 4·31-s − 5·32-s − 4·34-s − 4·35-s + 4·37-s − 4·38-s − 3·40-s + 2·41-s − 8·43-s − 8·46-s + 8·47-s + 9·49-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s − 0.447·5-s + 1.51·7-s + 1.06·8-s + 0.316·10-s − 1.06·14-s − 1/4·16-s + 0.970·17-s + 0.917·19-s + 0.223·20-s + 1.66·23-s + 1/5·25-s − 0.755·28-s + 1.48·29-s − 0.718·31-s − 0.883·32-s − 0.685·34-s − 0.676·35-s + 0.657·37-s − 0.648·38-s − 0.474·40-s + 0.312·41-s − 1.21·43-s − 1.17·46-s + 1.16·47-s + 9/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.657540739\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.657540739\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−7.962378739886146589380765359774, −7.45647039574327260208509109543, −6.83700283992453716964399581098, −5.43180465803786004508244291405, −5.13436160007850973775580214839, −4.41149586815688425709449109590, −3.62574452142788581674101283110, −2.58693416114083243677157112708, −1.31881299477957649819401512337, −0.875237802156576011489234191120,
0.875237802156576011489234191120, 1.31881299477957649819401512337, 2.58693416114083243677157112708, 3.62574452142788581674101283110, 4.41149586815688425709449109590, 5.13436160007850973775580214839, 5.43180465803786004508244291405, 6.83700283992453716964399581098, 7.45647039574327260208509109543, 7.962378739886146589380765359774
