| L(s) = 1 | − 2-s − 4-s − 2·5-s + 7-s + 3·8-s + 2·10-s + 11-s − 14-s − 16-s + 2·17-s + 4·19-s + 2·20-s − 22-s − 25-s − 28-s − 6·29-s + 4·31-s − 5·32-s − 2·34-s − 2·35-s − 6·37-s − 4·38-s − 6·40-s + 6·41-s + 12·43-s − 44-s + 4·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s − 0.894·5-s + 0.377·7-s + 1.06·8-s + 0.632·10-s + 0.301·11-s − 0.267·14-s − 1/4·16-s + 0.485·17-s + 0.917·19-s + 0.447·20-s − 0.213·22-s − 1/5·25-s − 0.188·28-s − 1.11·29-s + 0.718·31-s − 0.883·32-s − 0.342·34-s − 0.338·35-s − 0.986·37-s − 0.648·38-s − 0.948·40-s + 0.937·41-s + 1.82·43-s − 0.150·44-s + 0.583·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 117117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.166020198\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.166020198\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−13.59638442028590, −13.16475455354092, −12.52802014571226, −12.02734231117142, −11.66040070430642, −11.12063871191774, −10.65543244752087, −10.08424082103391, −9.680120652508632, −9.028679152082999, −8.772516110922545, −8.173437792138120, −7.594856664576335, −7.416995918147194, −6.935655066299235, −5.807227939354293, −5.637425901786496, −4.882380289254931, −4.171186559096408, −4.021695148205644, −3.301488885883444, −2.540564126887465, −1.688033911276760, −1.032372683482642, −0.4606856354816626,
0.4606856354816626, 1.032372683482642, 1.688033911276760, 2.540564126887465, 3.301488885883444, 4.021695148205644, 4.171186559096408, 4.882380289254931, 5.637425901786496, 5.807227939354293, 6.935655066299235, 7.416995918147194, 7.594856664576335, 8.173437792138120, 8.772516110922545, 9.028679152082999, 9.680120652508632, 10.08424082103391, 10.65543244752087, 11.12063871191774, 11.66040070430642, 12.02734231117142, 12.52802014571226, 13.16475455354092, 13.59638442028590
