L(s) = 1 | − 2-s + 4-s − 8-s + 11-s − 2·13-s + 16-s − 2·19-s − 22-s − 5·25-s + 2·26-s + 6·29-s − 2·31-s − 32-s + 2·37-s + 2·38-s − 4·43-s + 44-s − 6·47-s + 5·50-s − 2·52-s + 6·53-s − 6·58-s − 2·61-s + 2·62-s + 64-s − 4·67-s + 12·71-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s + 0.301·11-s − 0.554·13-s + 1/4·16-s − 0.458·19-s − 0.213·22-s − 25-s + 0.392·26-s + 1.11·29-s − 0.359·31-s − 0.176·32-s + 0.328·37-s + 0.324·38-s − 0.609·43-s + 0.150·44-s − 0.875·47-s + 0.707·50-s − 0.277·52-s + 0.824·53-s − 0.787·58-s − 0.256·61-s + 0.254·62-s + 1/8·64-s − 0.488·67-s + 1.42·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.146606821\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.146606821\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
good | 5 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−7.85753432302505928921103293660, −6.98338187441398347810360655747, −6.52121111519916052383579561807, −5.75791670905845557167098435741, −4.96022111668989750761323800786, −4.15948019503367063738020240961, −3.31912164781038622756964020494, −2.42588747445440968215441279751, −1.68581041078581093085939022337, −0.56363101267073554687128119537,
0.56363101267073554687128119537, 1.68581041078581093085939022337, 2.42588747445440968215441279751, 3.31912164781038622756964020494, 4.15948019503367063738020240961, 4.96022111668989750761323800786, 5.75791670905845557167098435741, 6.52121111519916052383579561807, 6.98338187441398347810360655747, 7.85753432302505928921103293660