L(s) = 1 | + 5-s + 4·11-s − 6·13-s + 2·17-s + 25-s + 6·29-s − 8·31-s + 10·37-s + 2·41-s − 4·43-s + 8·47-s − 2·53-s + 4·55-s + 8·59-s − 14·61-s − 6·65-s + 12·67-s + 16·71-s − 2·73-s − 8·79-s − 8·83-s + 2·85-s + 10·89-s − 2·97-s + 101-s + 103-s + 107-s + ⋯ |
L(s) = 1 | + 0.447·5-s + 1.20·11-s − 1.66·13-s + 0.485·17-s + 1/5·25-s + 1.11·29-s − 1.43·31-s + 1.64·37-s + 0.312·41-s − 0.609·43-s + 1.16·47-s − 0.274·53-s + 0.539·55-s + 1.04·59-s − 1.79·61-s − 0.744·65-s + 1.46·67-s + 1.89·71-s − 0.234·73-s − 0.900·79-s − 0.878·83-s + 0.216·85-s + 1.05·89-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 141120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 141120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.913998634\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.913998634\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−13.40300329809758, −12.79825365122501, −12.45885231993872, −12.01285752992712, −11.58691737968923, −11.00901566810824, −10.48474350491657, −9.916561457121067, −9.420331514986141, −9.347434338248213, −8.579278432037733, −7.987231577653324, −7.511906951364897, −6.880513908333487, −6.661688773522439, −5.835280669567660, −5.538965489054329, −4.787800102859473, −4.410637117227870, −3.763793274513808, −3.100620519938187, −2.470448622768015, −1.972886096080967, −1.191458099644019, −0.5385775362380189,
0.5385775362380189, 1.191458099644019, 1.972886096080967, 2.470448622768015, 3.100620519938187, 3.763793274513808, 4.410637117227870, 4.787800102859473, 5.538965489054329, 5.835280669567660, 6.661688773522439, 6.880513908333487, 7.511906951364897, 7.987231577653324, 8.579278432037733, 9.347434338248213, 9.420331514986141, 9.916561457121067, 10.48474350491657, 11.00901566810824, 11.58691737968923, 12.01285752992712, 12.45885231993872, 12.79825365122501, 13.40300329809758