L(s) = 1 | + 2-s + 4-s − 7-s + 8-s + 2·13-s − 14-s + 16-s − 6·17-s + 8·19-s + 4·23-s + 2·26-s − 28-s + 2·29-s + 8·31-s + 32-s − 6·34-s − 6·37-s + 8·38-s + 6·41-s + 8·43-s + 4·46-s + 4·47-s + 49-s + 2·52-s + 10·53-s − 56-s + 2·58-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.377·7-s + 0.353·8-s + 0.554·13-s − 0.267·14-s + 1/4·16-s − 1.45·17-s + 1.83·19-s + 0.834·23-s + 0.392·26-s − 0.188·28-s + 0.371·29-s + 1.43·31-s + 0.176·32-s − 1.02·34-s − 0.986·37-s + 1.29·38-s + 0.937·41-s + 1.21·43-s + 0.589·46-s + 0.583·47-s + 1/7·49-s + 0.277·52-s + 1.37·53-s − 0.133·56-s + 0.262·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 381150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 381150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(6.239575783\) |
\(L(\frac12)\) |
\(\approx\) |
\(6.239575783\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
good | 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 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
−12.54027345507539, −11.99074460402391, −11.65437690107066, −11.20940375291392, −10.77015389993223, −10.32475478976486, −9.787209216152055, −9.318876526876648, −8.799452527082729, −8.488539182077707, −7.770311753745321, −7.282808076577390, −6.884273047377451, −6.515490362690510, −5.823400087684193, −5.601178578979264, −4.872149923850692, −4.570455164671154, −3.879879937026035, −3.550599952345212, −2.751216625489746, −2.619481386905240, −1.827448393951353, −0.9765775661593282, −0.6740293286935815,
0.6740293286935815, 0.9765775661593282, 1.827448393951353, 2.619481386905240, 2.751216625489746, 3.550599952345212, 3.879879937026035, 4.570455164671154, 4.872149923850692, 5.601178578979264, 5.823400087684193, 6.515490362690510, 6.884273047377451, 7.282808076577390, 7.770311753745321, 8.488539182077707, 8.799452527082729, 9.318876526876648, 9.787209216152055, 10.32475478976486, 10.77015389993223, 11.20940375291392, 11.65437690107066, 11.99074460402391, 12.54027345507539