L(s) = 1 | + 2·5-s + 7-s − 2·13-s − 2·17-s + 4·19-s − 25-s + 6·29-s + 2·35-s − 6·37-s + 6·41-s + 8·43-s + 8·47-s + 49-s + 6·53-s + 12·59-s − 10·61-s − 4·65-s + 16·67-s − 8·71-s − 6·73-s − 8·79-s + 12·83-s − 4·85-s + 14·89-s − 2·91-s + 8·95-s − 6·97-s + ⋯ |
L(s) = 1 | + 0.894·5-s + 0.377·7-s − 0.554·13-s − 0.485·17-s + 0.917·19-s − 1/5·25-s + 1.11·29-s + 0.338·35-s − 0.986·37-s + 0.937·41-s + 1.21·43-s + 1.16·47-s + 1/7·49-s + 0.824·53-s + 1.56·59-s − 1.28·61-s − 0.496·65-s + 1.95·67-s − 0.949·71-s − 0.702·73-s − 0.900·79-s + 1.31·83-s − 0.433·85-s + 1.48·89-s − 0.209·91-s + 0.820·95-s − 0.609·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.405381429\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.405381429\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 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 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−8.554787423961038515377892584466, −7.60915127131745033749985180770, −7.04900248691071640708325829872, −6.12703308629561375850339965370, −5.49986762476306403273964369433, −4.78929916333548247562635941914, −3.90492431895471759544045387591, −2.73546974220382424094334233459, −2.06147913969900506083387610841, −0.909937067860099633286900303831,
0.909937067860099633286900303831, 2.06147913969900506083387610841, 2.73546974220382424094334233459, 3.90492431895471759544045387591, 4.78929916333548247562635941914, 5.49986762476306403273964369433, 6.12703308629561375850339965370, 7.04900248691071640708325829872, 7.60915127131745033749985180770, 8.554787423961038515377892584466