| L(s) = 1 | − 3-s − 5-s + 7-s + 9-s + 11-s + 15-s + 17-s + 19-s − 21-s − 23-s + 25-s − 27-s + 29-s + 31-s − 33-s − 35-s − 37-s − 41-s − 43-s − 45-s + 47-s + 49-s − 51-s + 53-s − 55-s − 57-s + 59-s + ⋯ |
| L(s) = 1 | − 3-s − 5-s + 7-s + 9-s + 11-s + 15-s + 17-s + 19-s − 21-s − 23-s + 25-s − 27-s + 29-s + 31-s − 33-s − 35-s − 37-s − 41-s − 43-s − 45-s + 47-s + 49-s − 51-s + 53-s − 55-s − 57-s + 59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 52 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 52 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.120399375\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.120399375\) |
| \(L(1)\) |
\(\approx\) |
\(0.8713210307\) |
| \(L(1)\) |
\(\approx\) |
\(0.8713210307\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 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
−33.39188429341471191966452020370, −32.130918944212022543707732212505, −30.65462354257998333077126465909, −29.96605086908258737691454882888, −28.36060067265987969920553739335, −27.5700553860226959624108044490, −26.80452166778121356797666299605, −24.77806886688149125010104651184, −23.85357173841631334662033338798, −22.89334998733978824074294694662, −21.77355875503196331906367868327, −20.38158636444838754787405658262, −18.97680756489750856132064601949, −17.78876592377285538515735922010, −16.65351704497698380774362422957, −15.50099658712472989934724763806, −14.12598376547782469184670531364, −12.01249686642698651628050571239, −11.696619311784069603920347762224, −10.19408728346746730037957304999, −8.2347222087365546193535963474, −6.93912337134673800967738606948, −5.23953945516230862748986000823, −3.940683154994453848546879015520, −1.09268043847536817058763036713,
1.09268043847536817058763036713, 3.940683154994453848546879015520, 5.23953945516230862748986000823, 6.93912337134673800967738606948, 8.2347222087365546193535963474, 10.19408728346746730037957304999, 11.696619311784069603920347762224, 12.01249686642698651628050571239, 14.12598376547782469184670531364, 15.50099658712472989934724763806, 16.65351704497698380774362422957, 17.78876592377285538515735922010, 18.97680756489750856132064601949, 20.38158636444838754787405658262, 21.77355875503196331906367868327, 22.89334998733978824074294694662, 23.85357173841631334662033338798, 24.77806886688149125010104651184, 26.80452166778121356797666299605, 27.5700553860226959624108044490, 28.36060067265987969920553739335, 29.96605086908258737691454882888, 30.65462354257998333077126465909, 32.130918944212022543707732212505, 33.39188429341471191966452020370
