| L(s) = 1 | + 3-s − 5-s − 0.274·7-s + 9-s − 4.75·11-s − 2.11·13-s − 15-s + 3.21·17-s − 1.31·19-s − 0.274·21-s − 4.76·23-s + 25-s + 27-s − 6.42·29-s + 3.56·31-s − 4.75·33-s + 0.274·35-s + 9.24·37-s − 2.11·39-s − 1.56·41-s + 2.64·43-s − 45-s + 7.55·47-s − 6.92·49-s + 3.21·51-s − 12.6·53-s + 4.75·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.103·7-s + 0.333·9-s − 1.43·11-s − 0.587·13-s − 0.258·15-s + 0.779·17-s − 0.302·19-s − 0.0598·21-s − 0.994·23-s + 0.200·25-s + 0.192·27-s − 1.19·29-s + 0.639·31-s − 0.827·33-s + 0.0463·35-s + 1.51·37-s − 0.339·39-s − 0.244·41-s + 0.404·43-s − 0.149·45-s + 1.10·47-s − 0.989·49-s + 0.450·51-s − 1.73·53-s + 0.640·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.610602413\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.610602413\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 67 | \( 1 - T \) |
| good | 7 | \( 1 + 0.274T + 7T^{2} \) |
| 11 | \( 1 + 4.75T + 11T^{2} \) |
| 13 | \( 1 + 2.11T + 13T^{2} \) |
| 17 | \( 1 - 3.21T + 17T^{2} \) |
| 19 | \( 1 + 1.31T + 19T^{2} \) |
| 23 | \( 1 + 4.76T + 23T^{2} \) |
| 29 | \( 1 + 6.42T + 29T^{2} \) |
| 31 | \( 1 - 3.56T + 31T^{2} \) |
| 37 | \( 1 - 9.24T + 37T^{2} \) |
| 41 | \( 1 + 1.56T + 41T^{2} \) |
| 43 | \( 1 - 2.64T + 43T^{2} \) |
| 47 | \( 1 - 7.55T + 47T^{2} \) |
| 53 | \( 1 + 12.6T + 53T^{2} \) |
| 59 | \( 1 - 11.9T + 59T^{2} \) |
| 61 | \( 1 - 5.61T + 61T^{2} \) |
| 71 | \( 1 - 5.03T + 71T^{2} \) |
| 73 | \( 1 - 0.416T + 73T^{2} \) |
| 79 | \( 1 - 15.6T + 79T^{2} \) |
| 83 | \( 1 + 8.57T + 83T^{2} \) |
| 89 | \( 1 - 6.86T + 89T^{2} \) |
| 97 | \( 1 + 11.2T + 97T^{2} \) |
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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
−7.83551250443032902505892552517, −7.46557446395586193514394019981, −6.50341709124031513203693346872, −5.68706066366431565166358202931, −4.99777877561630857729807129349, −4.22409000479842212363277377871, −3.44995415818261639306327940672, −2.66551215691499248762232404322, −2.00665248696187530211995344546, −0.58280721254064627998106802362,
0.58280721254064627998106802362, 2.00665248696187530211995344546, 2.66551215691499248762232404322, 3.44995415818261639306327940672, 4.22409000479842212363277377871, 4.99777877561630857729807129349, 5.68706066366431565166358202931, 6.50341709124031513203693346872, 7.46557446395586193514394019981, 7.83551250443032902505892552517
