| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1019 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1019 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.771089306\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.771089306\) |
| \(L(1)\) |
\(\approx\) |
\(1.279399371\) |
| \(L(1)\) |
\(\approx\) |
\(1.279399371\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 1019 | \( 1 \) |
| 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
−21.19010081816104732496912881550, −20.490725631470017364162749795629, −19.60149174654011328159419757772, −19.22632199589371535242538057245, −18.45826565064531113099861626157, −17.45348923086518899733870653280, −16.85183588877471472095270957926, −16.05101916384972194027047700457, −15.15213546909594004368631357995, −14.32496411753715826680046337400, −13.68140521627204442164342271470, −12.53294967708012715022014955872, −11.99861258364722083347070140697, −10.4708176175907577595817113525, −9.84542302404386980012657405827, −9.39957708875999268945747480685, −8.75142504279090724109370768874, −7.60513436802592665300744171861, −6.86664752909821252630413001151, −6.184193241070003455684712963014, −4.90803429362631479239316542215, −3.23488469956398747216060905318, −2.9109963473712179693133787343, −1.7143077972074366534448567163, −0.90180606425925583122078903008,
0.90180606425925583122078903008, 1.7143077972074366534448567163, 2.9109963473712179693133787343, 3.23488469956398747216060905318, 4.90803429362631479239316542215, 6.184193241070003455684712963014, 6.86664752909821252630413001151, 7.60513436802592665300744171861, 8.75142504279090724109370768874, 9.39957708875999268945747480685, 9.84542302404386980012657405827, 10.4708176175907577595817113525, 11.99861258364722083347070140697, 12.53294967708012715022014955872, 13.68140521627204442164342271470, 14.32496411753715826680046337400, 15.15213546909594004368631357995, 16.05101916384972194027047700457, 16.85183588877471472095270957926, 17.45348923086518899733870653280, 18.45826565064531113099861626157, 19.22632199589371535242538057245, 19.60149174654011328159419757772, 20.490725631470017364162749795629, 21.19010081816104732496912881550
