| L(s) = 1 | − 2-s + 4-s + 2·5-s − 8-s − 2·10-s − 11-s − 4·13-s + 16-s + 6·17-s − 2·19-s + 2·20-s + 22-s − 25-s + 4·26-s + 6·29-s − 2·31-s − 32-s − 6·34-s + 2·37-s + 2·38-s − 2·40-s − 2·41-s − 12·43-s − 44-s − 6·47-s + 50-s − 4·52-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 − 1.10·13-s + 1/4·16-s + 1.45·17-s − 0.458·19-s + 0.447·20-s + 0.213·22-s − 1/5·25-s + 0.784·26-s + 1.11·29-s − 0.359·31-s − 0.176·32-s − 1.02·34-s + 0.328·37-s + 0.324·38-s − 0.316·40-s − 0.312·41-s − 1.82·43-s − 0.150·44-s − 0.875·47-s + 0.141·50-s − 0.554·52-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 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
−7.46758067332539910168084752489, −6.64844043492678762667548571037, −6.13947703686177335991368807841, −5.26964445931964368989458250501, −4.85404470743878570796267057581, −3.59335919373833731278220557054, −2.81163540902542034522408294565, −2.06347451200798813330821105283, −1.26397067800308264554359809769, 0,
1.26397067800308264554359809769, 2.06347451200798813330821105283, 2.81163540902542034522408294565, 3.59335919373833731278220557054, 4.85404470743878570796267057581, 5.26964445931964368989458250501, 6.13947703686177335991368807841, 6.64844043492678762667548571037, 7.46758067332539910168084752489
