L(s) = 1 | + 2-s − 4-s − 7-s − 3·8-s + 4·13-s − 14-s − 16-s + 2·17-s + 4·26-s + 28-s + 8·29-s − 4·31-s + 5·32-s + 2·34-s + 8·37-s + 4·41-s + 8·43-s − 12·47-s + 49-s − 4·52-s + 6·53-s + 3·56-s + 8·58-s + 8·59-s + 10·61-s − 4·62-s + 7·64-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1/2·4-s − 0.377·7-s − 1.06·8-s + 1.10·13-s − 0.267·14-s − 1/4·16-s + 0.485·17-s + 0.784·26-s + 0.188·28-s + 1.48·29-s − 0.718·31-s + 0.883·32-s + 0.342·34-s + 1.31·37-s + 0.624·41-s + 1.21·43-s − 1.75·47-s + 1/7·49-s − 0.554·52-s + 0.824·53-s + 0.400·56-s + 1.05·58-s + 1.04·59-s + 1.28·61-s − 0.508·62-s + 7/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.947504912\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.947504912\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 2 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 - 12 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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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
−9.424164907662773330689773273867, −8.631216710882126825023655008830, −7.967735853348289295528215854361, −6.75009946098867107228009858468, −6.02682507567955722890216660485, −5.31545457379330910918547105406, −4.29562056602394841199154433748, −3.60721839203858358057857219618, −2.66224727605981919951627179868, −0.911567385176684330942671980760,
0.911567385176684330942671980760, 2.66224727605981919951627179868, 3.60721839203858358057857219618, 4.29562056602394841199154433748, 5.31545457379330910918547105406, 6.02682507567955722890216660485, 6.75009946098867107228009858468, 7.967735853348289295528215854361, 8.631216710882126825023655008830, 9.424164907662773330689773273867