L(s) = 1 | − 1.61·2-s + 1.61·4-s + 0.618·7-s − 8-s + 9-s − 1.61·13-s − 1.00·14-s + 0.618·17-s − 1.61·18-s − 1.61·19-s − 1.61·23-s + 25-s + 2.61·26-s + 1.00·28-s + 0.618·29-s + 32-s − 1.00·34-s + 1.61·36-s + 2.61·38-s − 1.61·41-s + 2.61·46-s − 0.618·49-s − 1.61·50-s − 2.61·52-s − 0.618·56-s − 1.00·58-s + 0.618·59-s + ⋯ |
L(s) = 1 | − 1.61·2-s + 1.61·4-s + 0.618·7-s − 8-s + 9-s − 1.61·13-s − 1.00·14-s + 0.618·17-s − 1.61·18-s − 1.61·19-s − 1.61·23-s + 25-s + 2.61·26-s + 1.00·28-s + 0.618·29-s + 32-s − 1.00·34-s + 1.61·36-s + 2.61·38-s − 1.61·41-s + 2.61·46-s − 0.618·49-s − 1.61·50-s − 2.61·52-s − 0.618·56-s − 1.00·58-s + 0.618·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3101772644\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3101772644\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 103 | \( 1 - T \) |
good | 2 | \( 1 + 1.61T + T^{2} \) |
| 3 | \( 1 - T^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 7 | \( 1 - 0.618T + T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 + 1.61T + T^{2} \) |
| 17 | \( 1 - 0.618T + T^{2} \) |
| 19 | \( 1 + 1.61T + T^{2} \) |
| 23 | \( 1 + 1.61T + T^{2} \) |
| 29 | \( 1 - 0.618T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + 1.61T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - 0.618T + T^{2} \) |
| 61 | \( 1 - 0.618T + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + 1.61T + T^{2} \) |
| 83 | \( 1 - 0.618T + T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - 0.618T + T^{2} \) |
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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
−14.35488539200342705454252134633, −12.70894104994314844385392810633, −11.72604188680177212338039429439, −10.34214505526155332385890590804, −9.959707971151309180493693391976, −8.593356723596807581237004373057, −7.69809741917201771363481039022, −6.71017783955903441438614047649, −4.64753948580894546247106478607, −2.00946183109297238095017317494,
2.00946183109297238095017317494, 4.64753948580894546247106478607, 6.71017783955903441438614047649, 7.69809741917201771363481039022, 8.593356723596807581237004373057, 9.959707971151309180493693391976, 10.34214505526155332385890590804, 11.72604188680177212338039429439, 12.70894104994314844385392810633, 14.35488539200342705454252134633