L(s) = 1 | + 3-s + 5-s + 9-s + 4·11-s − 2·13-s + 15-s + 2·17-s − 19-s + 25-s + 27-s + 6·29-s + 4·33-s + 6·37-s − 2·39-s − 6·41-s + 4·43-s + 45-s + 8·47-s − 7·49-s + 2·51-s + 6·53-s + 4·55-s − 57-s − 12·59-s − 2·61-s − 2·65-s + 4·67-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1/3·9-s + 1.20·11-s − 0.554·13-s + 0.258·15-s + 0.485·17-s − 0.229·19-s + 1/5·25-s + 0.192·27-s + 1.11·29-s + 0.696·33-s + 0.986·37-s − 0.320·39-s − 0.937·41-s + 0.609·43-s + 0.149·45-s + 1.16·47-s − 49-s + 0.280·51-s + 0.824·53-s + 0.539·55-s − 0.132·57-s − 1.56·59-s − 0.256·61-s − 0.248·65-s + 0.488·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.647386668\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.647386668\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 19 | \( 1 + T \) |
good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−9.085138980992414803706980641781, −8.329380367409197432933017254307, −7.50626171816380600278299971037, −6.68414571793939990988793717525, −6.02487406501461713440146826459, −4.93565977359749539952919191174, −4.12452225722897632118267519001, −3.16783272905183981738929591434, −2.20529880534869042323299350797, −1.10597384283023648339975141407,
1.10597384283023648339975141407, 2.20529880534869042323299350797, 3.16783272905183981738929591434, 4.12452225722897632118267519001, 4.93565977359749539952919191174, 6.02487406501461713440146826459, 6.68414571793939990988793717525, 7.50626171816380600278299971037, 8.329380367409197432933017254307, 9.085138980992414803706980641781