| L(s) = 1 | + 3-s − 5-s − 7-s + 9-s − 11-s + 13-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 + 39-s − 41-s − 43-s − 45-s + 47-s + 49-s + 51-s + 53-s + 55-s + ⋯ |
| L(s) = 1 | + 3-s − 5-s − 7-s + 9-s − 11-s + 13-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 + 39-s − 41-s − 43-s − 45-s + 47-s + 49-s + 51-s + 53-s + 55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1208 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1208 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.800117077\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.800117077\) |
| \(L(1)\) |
\(\approx\) |
\(1.084670057\) |
| \(L(1)\) |
\(\approx\) |
\(1.084670057\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 151 | \( 1 \) |
| good | 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
−20.75375898153694144385686374340, −20.241947518978322012828470260063, −19.38108556471342586056163876776, −18.76725862033597481902646214459, −18.42378538568075817074385385961, −16.90256791776217161255811721405, −16.048739070502712135552280886382, −15.6055441585291427380763068882, −14.971311602418016263438986244427, −13.923422898467260153541241990940, −13.196026733769497622283358899298, −12.57540290617251053233608571626, −11.72515106967013135733080481087, −10.44380306270033678737754220407, −10.08955824747449923686096593361, −8.83586725047140613907328067761, −8.31569418251284153839433829161, −7.550699143749775514644458187694, −6.735444861521432109488562150186, −5.681370697472833741159133837156, −4.390066390023535015453115644143, −3.584150460376705638262974716921, −3.080809328457613285790691080579, −1.94132641186625724406587146156, −0.55399561308367406809073168305,
0.55399561308367406809073168305, 1.94132641186625724406587146156, 3.080809328457613285790691080579, 3.584150460376705638262974716921, 4.390066390023535015453115644143, 5.681370697472833741159133837156, 6.735444861521432109488562150186, 7.550699143749775514644458187694, 8.31569418251284153839433829161, 8.83586725047140613907328067761, 10.08955824747449923686096593361, 10.44380306270033678737754220407, 11.72515106967013135733080481087, 12.57540290617251053233608571626, 13.196026733769497622283358899298, 13.923422898467260153541241990940, 14.971311602418016263438986244427, 15.6055441585291427380763068882, 16.048739070502712135552280886382, 16.90256791776217161255811721405, 18.42378538568075817074385385961, 18.76725862033597481902646214459, 19.38108556471342586056163876776, 20.241947518978322012828470260063, 20.75375898153694144385686374340
