| L(s) = 1 | − 2-s + 4-s + 2·5-s − 8-s − 2·10-s − 11-s + 2·13-s + 16-s + 6·17-s + 4·19-s + 2·20-s + 22-s − 6·23-s − 25-s − 2·26-s − 6·29-s − 2·31-s − 32-s − 6·34-s + 2·37-s − 4·38-s − 2·40-s + 10·41-s − 6·43-s − 44-s + 6·46-s + 50-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.894·5-s − 0.353·8-s − 0.632·10-s − 0.301·11-s + 0.554·13-s + 1/4·16-s + 1.45·17-s + 0.917·19-s + 0.447·20-s + 0.213·22-s − 1.25·23-s − 1/5·25-s − 0.392·26-s − 1.11·29-s − 0.359·31-s − 0.176·32-s − 1.02·34-s + 0.328·37-s − 0.648·38-s − 0.316·40-s + 1.56·41-s − 0.914·43-s − 0.150·44-s + 0.884·46-s + 0.141·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.881077140\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.881077140\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−7.921830767903585439800511403218, −7.09838797028536757056000094974, −6.29495030300597056567800873515, −5.58803284371119489092612791896, −5.37607448225939459833443134024, −3.99891936171589168831979616858, −3.34460554432934150570545857388, −2.36473181899141159319876015506, −1.66572154457764755696544381865, −0.74856356984791146838929512524,
0.74856356984791146838929512524, 1.66572154457764755696544381865, 2.36473181899141159319876015506, 3.34460554432934150570545857388, 3.99891936171589168831979616858, 5.37607448225939459833443134024, 5.58803284371119489092612791896, 6.29495030300597056567800873515, 7.09838797028536757056000094974, 7.921830767903585439800511403218
