| L(s) = 1 | − 2·5-s + 7-s + 5·13-s − 8·17-s + 3·19-s + 23-s − 25-s + 8·29-s + 8·31-s − 2·35-s − 2·37-s + 2·41-s + 2·43-s − 4·47-s + 49-s − 12·53-s − 15·59-s + 10·61-s − 10·65-s − 8·71-s + 4·73-s + 3·79-s + 3·83-s + 16·85-s − 6·89-s + 5·91-s − 6·95-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.377·7-s + 1.38·13-s − 1.94·17-s + 0.688·19-s + 0.208·23-s − 1/5·25-s + 1.48·29-s + 1.43·31-s − 0.338·35-s − 0.328·37-s + 0.312·41-s + 0.304·43-s − 0.583·47-s + 1/7·49-s − 1.64·53-s − 1.95·59-s + 1.28·61-s − 1.24·65-s − 0.949·71-s + 0.468·73-s + 0.337·79-s + 0.329·83-s + 1.73·85-s − 0.635·89-s + 0.524·91-s − 0.615·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 121968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121968 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + 8 T + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 15 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 3 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−13.82207126456680, −13.40063817480703, −12.82587122131826, −12.29742750540220, −11.67077187857675, −11.46680838409334, −10.94998371500734, −10.58531248874892, −9.954797025812135, −9.224003698115006, −8.870675882117600, −8.239643619704183, −8.077961217036304, −7.433184963120555, −6.687069270595192, −6.434478157852472, −5.895312140840040, −5.006215238992062, −4.543609912043591, −4.225692834062723, −3.483114039800328, −2.998007630471681, −2.283989376258810, −1.477761602560985, −0.8632074000131349, 0,
0.8632074000131349, 1.477761602560985, 2.283989376258810, 2.998007630471681, 3.483114039800328, 4.225692834062723, 4.543609912043591, 5.006215238992062, 5.895312140840040, 6.434478157852472, 6.687069270595192, 7.433184963120555, 8.077961217036304, 8.239643619704183, 8.870675882117600, 9.224003698115006, 9.954797025812135, 10.58531248874892, 10.94998371500734, 11.46680838409334, 11.67077187857675, 12.29742750540220, 12.82587122131826, 13.40063817480703, 13.82207126456680
