L(s) = 1 | + 0.618·2-s − 0.618·4-s − 1.61·7-s − 8-s + 9-s + 0.618·13-s − 1.00·14-s − 1.61·17-s + 0.618·18-s + 0.618·19-s + 0.618·23-s + 25-s + 0.381·26-s + 0.999·28-s − 1.61·29-s + 0.999·32-s − 1.00·34-s − 0.618·36-s + 0.381·38-s + 0.618·41-s + 0.381·46-s + 1.61·49-s + 0.618·50-s − 0.381·52-s + 1.61·56-s − 1.00·58-s − 1.61·59-s + ⋯ |
L(s) = 1 | + 0.618·2-s − 0.618·4-s − 1.61·7-s − 8-s + 9-s + 0.618·13-s − 1.00·14-s − 1.61·17-s + 0.618·18-s + 0.618·19-s + 0.618·23-s + 25-s + 0.381·26-s + 0.999·28-s − 1.61·29-s + 0.999·32-s − 1.00·34-s − 0.618·36-s + 0.381·38-s + 0.618·41-s + 0.381·46-s + 1.61·49-s + 0.618·50-s − 0.381·52-s + 1.61·56-s − 1.00·58-s − 1.61·59-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
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6119819366\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6119819366\) |
\(L(1)\) |
|
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 | \( 1 - 0.618T + T^{2} \) |
| 3 | \( 1 - T^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 7 | \( 1 + 1.61T + T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 - 0.618T + T^{2} \) |
| 17 | \( 1 + 1.61T + T^{2} \) |
| 19 | \( 1 - 0.618T + T^{2} \) |
| 23 | \( 1 - 0.618T + T^{2} \) |
| 29 | \( 1 + 1.61T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - 0.618T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + 1.61T + T^{2} \) |
| 61 | \( 1 + 1.61T + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - 0.618T + T^{2} \) |
| 83 | \( 1 + 1.61T + T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + 1.61T + T^{2} \) |
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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
−13.68380576693065448866038657735, −13.06999783634067273401280863882, −12.51791807552834908624905200911, −10.91161638857600902781983446815, −9.587760247432759093520355809873, −8.974746013144302001889982996005, −7.07266036729859331897519459089, −6.03454633619783779298064033402, −4.46895357445124374556423870030, −3.28396214040166531134869327212,
3.28396214040166531134869327212, 4.46895357445124374556423870030, 6.03454633619783779298064033402, 7.07266036729859331897519459089, 8.974746013144302001889982996005, 9.587760247432759093520355809873, 10.91161638857600902781983446815, 12.51791807552834908624905200911, 13.06999783634067273401280863882, 13.68380576693065448866038657735