L(s) = 1 | − 6·11-s − 13-s + 4·17-s + 6·19-s + 6·23-s + 2·29-s + 6·37-s − 2·41-s + 12·43-s + 12·47-s − 7·49-s + 10·53-s − 6·59-s + 10·61-s + 12·67-s − 12·71-s − 2·73-s + 12·83-s + 10·89-s + 10·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
L(s) = 1 | − 1.80·11-s − 0.277·13-s + 0.970·17-s + 1.37·19-s + 1.25·23-s + 0.371·29-s + 0.986·37-s − 0.312·41-s + 1.82·43-s + 1.75·47-s − 49-s + 1.37·53-s − 0.781·59-s + 1.28·61-s + 1.46·67-s − 1.42·71-s − 0.234·73-s + 1.31·83-s + 1.05·89-s + 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 93600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.834069548\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.834069548\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−13.62881210992517, −13.43646993088189, −12.84066631289788, −12.38329935899019, −11.95671222205450, −11.28709646446297, −10.85801986983373, −10.32228418754987, −9.967769009120792, −9.348492722741769, −8.921188472028757, −8.161082889176266, −7.677649668083378, −7.450592241212141, −6.861939782290933, −6.013055017660177, −5.431580403464993, −5.248429878011646, −4.594777422843996, −3.852458879294106, −3.075526375477105, −2.753679476981630, −2.156771050461284, −1.025257021550912, −0.6442035328057621,
0.6442035328057621, 1.025257021550912, 2.156771050461284, 2.753679476981630, 3.075526375477105, 3.852458879294106, 4.594777422843996, 5.248429878011646, 5.431580403464993, 6.013055017660177, 6.861939782290933, 7.450592241212141, 7.677649668083378, 8.161082889176266, 8.921188472028757, 9.348492722741769, 9.967769009120792, 10.32228418754987, 10.85801986983373, 11.28709646446297, 11.95671222205450, 12.38329935899019, 12.84066631289788, 13.43646993088189, 13.62881210992517