| L(s) = 1 | − 3-s + 2·5-s + 7-s − 2·9-s + 3·11-s + 3·13-s − 2·15-s − 2·17-s − 21-s − 3·23-s − 25-s + 5·27-s + 6·29-s + 4·31-s − 3·33-s + 2·35-s + 8·37-s − 3·39-s − 12·41-s + 7·43-s − 4·45-s + 3·47-s − 6·49-s + 2·51-s + 12·53-s + 6·55-s + 8·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s + 0.377·7-s − 2/3·9-s + 0.904·11-s + 0.832·13-s − 0.516·15-s − 0.485·17-s − 0.218·21-s − 0.625·23-s − 1/5·25-s + 0.962·27-s + 1.11·29-s + 0.718·31-s − 0.522·33-s + 0.338·35-s + 1.31·37-s − 0.480·39-s − 1.87·41-s + 1.06·43-s − 0.596·45-s + 0.437·47-s − 6/7·49-s + 0.280·51-s + 1.64·53-s + 0.809·55-s + 1.04·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.986891683\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.986891683\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 503 | \( 1 - T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 + 9 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 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
−8.655303215003303634028338901122, −7.76645744313135039888829009886, −6.62250615857048609982420142518, −6.21875799719562321954199448070, −5.65471001098464590834080164831, −4.76384903599026596609320353409, −3.96925302067654004545448729937, −2.84856353531718499178498954549, −1.87499163843732066730306709518, −0.861579764966600287825408224277,
0.861579764966600287825408224277, 1.87499163843732066730306709518, 2.84856353531718499178498954549, 3.96925302067654004545448729937, 4.76384903599026596609320353409, 5.65471001098464590834080164831, 6.21875799719562321954199448070, 6.62250615857048609982420142518, 7.76645744313135039888829009886, 8.655303215003303634028338901122
