L(s) = 1 | − 2·2-s − 3-s + 2·4-s + 2·6-s + 2·7-s + 9-s − 2·12-s − 4·14-s − 4·16-s − 17-s − 2·18-s + 8·19-s − 2·21-s + 6·23-s − 27-s + 4·28-s + 3·29-s − 11·31-s + 8·32-s + 2·34-s + 2·36-s − 6·37-s − 16·38-s + 4·42-s − 12·46-s − 2·47-s + 4·48-s + ⋯ |
L(s) = 1 | − 1.41·2-s − 0.577·3-s + 4-s + 0.816·6-s + 0.755·7-s + 1/3·9-s − 0.577·12-s − 1.06·14-s − 16-s − 0.242·17-s − 0.471·18-s + 1.83·19-s − 0.436·21-s + 1.25·23-s − 0.192·27-s + 0.755·28-s + 0.557·29-s − 1.97·31-s + 1.41·32-s + 0.342·34-s + 1/3·36-s − 0.986·37-s − 2.59·38-s + 0.617·42-s − 1.76·46-s − 0.291·47-s + 0.577·48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 215475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 215475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.354697800\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.354697800\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 13 | \( 1 \) |
| 17 | \( 1 + T \) |
good | 2 | \( 1 + p T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 11 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 11 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−12.96895173505937, −12.31414653885548, −11.84588680377580, −11.38324044919957, −11.07523345227071, −10.62337886230655, −10.22790871222339, −9.618784604497219, −9.152073108712423, −8.968655745139370, −8.218308490381263, −7.840277171739862, −7.349264726065248, −6.955623382081368, −6.572025730612780, −5.675271735793247, −5.167376368857252, −4.979413018821758, −4.213723110285109, −3.491384272695407, −2.949855568202891, −1.954253996116994, −1.721316375657183, −0.8773279770723375, −0.5710481256047217,
0.5710481256047217, 0.8773279770723375, 1.721316375657183, 1.954253996116994, 2.949855568202891, 3.491384272695407, 4.213723110285109, 4.979413018821758, 5.167376368857252, 5.675271735793247, 6.572025730612780, 6.955623382081368, 7.349264726065248, 7.840277171739862, 8.218308490381263, 8.968655745139370, 9.152073108712423, 9.618784604497219, 10.22790871222339, 10.62337886230655, 11.07523345227071, 11.38324044919957, 11.84588680377580, 12.31414653885548, 12.96895173505937