| L(s) = 1 | + 2-s − 4-s + 7-s − 3·8-s − 3·9-s − 4·11-s + 2·13-s + 14-s − 16-s − 6·17-s − 3·18-s + 4·19-s − 4·22-s − 23-s + 2·26-s − 28-s − 2·29-s + 4·31-s + 5·32-s − 6·34-s + 3·36-s + 10·37-s + 4·38-s + 10·41-s − 12·43-s + 4·44-s − 46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s + 0.377·7-s − 1.06·8-s − 9-s − 1.20·11-s + 0.554·13-s + 0.267·14-s − 1/4·16-s − 1.45·17-s − 0.707·18-s + 0.917·19-s − 0.852·22-s − 0.208·23-s + 0.392·26-s − 0.188·28-s − 0.371·29-s + 0.718·31-s + 0.883·32-s − 1.02·34-s + 1/2·36-s + 1.64·37-s + 0.648·38-s + 1.56·41-s − 1.82·43-s + 0.603·44-s − 0.147·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.544988028\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.544988028\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 3 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−8.257751522950231816488875233294, −8.016124128018196443937064440201, −6.80495531146416711043233780876, −5.94221101431227801078388923498, −5.42521227608983565445028389193, −4.71901779676615522584981056956, −3.97527841357547486453795945979, −2.97933848611012499979092091966, −2.34495592904849279164106500380, −0.61612818314657448403323002463,
0.61612818314657448403323002463, 2.34495592904849279164106500380, 2.97933848611012499979092091966, 3.97527841357547486453795945979, 4.71901779676615522584981056956, 5.42521227608983565445028389193, 5.94221101431227801078388923498, 6.80495531146416711043233780876, 8.016124128018196443937064440201, 8.257751522950231816488875233294
