| L(s) = 1 | − 3-s + 5-s − 5·7-s + 9-s + 5·11-s + 13-s − 15-s − 5·17-s + 3·19-s + 5·21-s + 3·23-s + 25-s − 27-s − 6·29-s − 6·31-s − 5·33-s − 5·35-s + 37-s − 39-s − 4·43-s + 45-s + 18·49-s + 5·51-s + 3·53-s + 5·55-s − 3·57-s + 10·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 1.88·7-s + 1/3·9-s + 1.50·11-s + 0.277·13-s − 0.258·15-s − 1.21·17-s + 0.688·19-s + 1.09·21-s + 0.625·23-s + 1/5·25-s − 0.192·27-s − 1.11·29-s − 1.07·31-s − 0.870·33-s − 0.845·35-s + 0.164·37-s − 0.160·39-s − 0.609·43-s + 0.149·45-s + 18/7·49-s + 0.700·51-s + 0.412·53-s + 0.674·55-s − 0.397·57-s + 1.30·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 37 | \( 1 - T \) |
| good | 7 | \( 1 + 5 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 9 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 5 T + p T^{2} \) |
| 89 | \( 1 - 13 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−15.21694723556411, −14.74331700739874, −14.05594384613860, −13.38894332422525, −13.13543861830404, −12.68352259169699, −12.01737115120917, −11.55639552234000, −10.95159752486898, −10.45821732114205, −9.691411800540682, −9.247695077401959, −9.205793712036249, −8.327524545007509, −7.205323610803177, −6.866317010482658, −6.548739227170847, −5.860145812745538, −5.495936506614535, −4.551546191195818, −3.770047100909863, −3.508084202590655, −2.599388878254291, −1.761051470212527, −0.8911441472596030, 0,
0.8911441472596030, 1.761051470212527, 2.599388878254291, 3.508084202590655, 3.770047100909863, 4.551546191195818, 5.495936506614535, 5.860145812745538, 6.548739227170847, 6.866317010482658, 7.205323610803177, 8.327524545007509, 9.205793712036249, 9.247695077401959, 9.691411800540682, 10.45821732114205, 10.95159752486898, 11.55639552234000, 12.01737115120917, 12.68352259169699, 13.13543861830404, 13.38894332422525, 14.05594384613860, 14.74331700739874, 15.21694723556411
