L(s) = 1 | + 3-s − 5-s − 7-s + 9-s + 2·13-s − 15-s − 2·17-s + 4·19-s − 21-s + 23-s + 25-s + 27-s − 10·29-s + 4·31-s + 35-s − 6·37-s + 2·39-s + 2·41-s + 8·43-s − 45-s − 12·47-s + 49-s − 2·51-s − 6·53-s + 4·57-s − 12·59-s + 2·61-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s + 0.554·13-s − 0.258·15-s − 0.485·17-s + 0.917·19-s − 0.218·21-s + 0.208·23-s + 1/5·25-s + 0.192·27-s − 1.85·29-s + 0.718·31-s + 0.169·35-s − 0.986·37-s + 0.320·39-s + 0.312·41-s + 1.21·43-s − 0.149·45-s − 1.75·47-s + 1/7·49-s − 0.280·51-s − 0.824·53-s + 0.529·57-s − 1.56·59-s + 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.157466346\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.157466346\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−14.86842182556933, −14.30141828330864, −13.69622413838758, −13.41784826229874, −12.68146158375870, −12.41359726138753, −11.60726238654523, −11.13812802375031, −10.73622406648993, −9.921992273947172, −9.442823128446419, −9.028360037405768, −8.425973265889322, −7.753902679054633, −7.462251474780715, −6.691660386964164, −6.222179157905945, −5.442696341829791, −4.820493476328567, −4.115233510216174, −3.451241599506593, −3.133702793658873, −2.194761410400353, −1.493332709133943, −0.5240716158634332,
0.5240716158634332, 1.493332709133943, 2.194761410400353, 3.133702793658873, 3.451241599506593, 4.115233510216174, 4.820493476328567, 5.442696341829791, 6.222179157905945, 6.691660386964164, 7.462251474780715, 7.753902679054633, 8.425973265889322, 9.028360037405768, 9.442823128446419, 9.921992273947172, 10.73622406648993, 11.13812802375031, 11.60726238654523, 12.41359726138753, 12.68146158375870, 13.41784826229874, 13.69622413838758, 14.30141828330864, 14.86842182556933