L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s + 7-s − 8-s + 9-s − 10-s − 11-s + 12-s + 13-s − 14-s + 15-s + 16-s − 17-s − 18-s − 19-s + 20-s + 21-s + 22-s + 23-s − 24-s + 25-s − 26-s + 27-s + 28-s + ⋯ |
L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s + 7-s − 8-s + 9-s − 10-s − 11-s + 12-s + 13-s − 14-s + 15-s + 16-s − 17-s − 18-s − 19-s + 20-s + 21-s + 22-s + 23-s − 24-s + 25-s − 26-s + 27-s + 28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 419 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 419 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.726129511\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.726129511\) |
\(L(1)\) |
\(\approx\) |
\(1.381291599\) |
\(L(1)\) |
\(\approx\) |
\(1.381291599\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 419 | \( 1 \) |
good | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| 37 | \( 1 + T \) |
| 41 | \( 1 + T \) |
| 43 | \( 1 + T \) |
| 47 | \( 1 + T \) |
| 53 | \( 1 - T \) |
| 59 | \( 1 + T \) |
| 61 | \( 1 - T \) |
| 67 | \( 1 - T \) |
| 71 | \( 1 - T \) |
| 73 | \( 1 + T \) |
| 79 | \( 1 + T \) |
| 83 | \( 1 - T \) |
| 89 | \( 1 - T \) |
| 97 | \( 1 + T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−24.284005557180463673177750092236, −23.59991986941719654124635127150, −21.74173284027447897314993654040, −20.964458544698368508897783318021, −20.76005016737446952923392486423, −19.66729936561195079783412960121, −18.595814318703163578469084574285, −18.07667759961316780571663801138, −17.33835620814682973152425457277, −16.13403524521494957231592263686, −15.266960337777193408520644949540, −14.48015986379768539146042678004, −13.43666788930377851317881739633, −12.6633797826727061403380517540, −10.876806805115837511376247367331, −10.67961657540470984865912612726, −9.32713513139667989058197898434, −8.72350533364957274307745017998, −7.9609634965533502906033284472, −6.923666381524386323244413387831, −5.81307400719492880729080547726, −4.44283222007722139186533808840, −2.78147512537896162074161363340, −2.09923184485817927515323486002, −1.09856054097616453791438539345,
1.09856054097616453791438539345, 2.09923184485817927515323486002, 2.78147512537896162074161363340, 4.44283222007722139186533808840, 5.81307400719492880729080547726, 6.923666381524386323244413387831, 7.9609634965533502906033284472, 8.72350533364957274307745017998, 9.32713513139667989058197898434, 10.67961657540470984865912612726, 10.876806805115837511376247367331, 12.6633797826727061403380517540, 13.43666788930377851317881739633, 14.48015986379768539146042678004, 15.266960337777193408520644949540, 16.13403524521494957231592263686, 17.33835620814682973152425457277, 18.07667759961316780571663801138, 18.595814318703163578469084574285, 19.66729936561195079783412960121, 20.76005016737446952923392486423, 20.964458544698368508897783318021, 21.74173284027447897314993654040, 23.59991986941719654124635127150, 24.284005557180463673177750092236