| L(s) = 1 | + 3-s − 5-s − 0.184·7-s + 9-s + 3.61·11-s − 1.46·13-s − 15-s − 5.78·17-s − 2.96·19-s − 0.184·21-s − 3·23-s + 25-s + 27-s − 0.815·29-s − 4.89·31-s + 3.61·33-s + 0.184·35-s + 2.63·37-s − 1.46·39-s − 0.0338·41-s + 6.39·43-s − 45-s − 8.66·47-s − 6.96·49-s − 5.78·51-s + 3.89·53-s − 3.61·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.0695·7-s + 0.333·9-s + 1.09·11-s − 0.406·13-s − 0.258·15-s − 1.40·17-s − 0.680·19-s − 0.0401·21-s − 0.625·23-s + 0.200·25-s + 0.192·27-s − 0.151·29-s − 0.879·31-s + 0.629·33-s + 0.0311·35-s + 0.432·37-s − 0.234·39-s − 0.00529·41-s + 0.975·43-s − 0.149·45-s − 1.26·47-s − 0.995·49-s − 0.809·51-s + 0.535·53-s − 0.487·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.184T + 7T^{2} \) |
| 11 | \( 1 - 3.61T + 11T^{2} \) |
| 13 | \( 1 + 1.46T + 13T^{2} \) |
| 17 | \( 1 + 5.78T + 17T^{2} \) |
| 19 | \( 1 + 2.96T + 19T^{2} \) |
| 23 | \( 1 + 3T + 23T^{2} \) |
| 29 | \( 1 + 0.815T + 29T^{2} \) |
| 31 | \( 1 + 4.89T + 31T^{2} \) |
| 37 | \( 1 - 2.63T + 37T^{2} \) |
| 41 | \( 1 + 0.0338T + 41T^{2} \) |
| 43 | \( 1 - 6.39T + 43T^{2} \) |
| 47 | \( 1 + 8.66T + 47T^{2} \) |
| 53 | \( 1 - 3.89T + 53T^{2} \) |
| 59 | \( 1 + 7.26T + 59T^{2} \) |
| 61 | \( 1 + 10.9T + 61T^{2} \) |
| 71 | \( 1 + 1.86T + 71T^{2} \) |
| 73 | \( 1 - 5.45T + 73T^{2} \) |
| 79 | \( 1 + 4.46T + 79T^{2} \) |
| 83 | \( 1 + 9.92T + 83T^{2} \) |
| 89 | \( 1 - 16.7T + 89T^{2} \) |
| 97 | \( 1 - 8.37T + 97T^{2} \) |
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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
−8.087879855501762398758829369040, −7.41804175913356590854194182500, −6.63695554534858477095513856212, −6.08874624632761653353625403548, −4.79613870249856371703601441021, −4.20702906396356179995133393907, −3.50456003493790340081758617816, −2.45244224726155159119614850906, −1.58123603926643528750994292919, 0,
1.58123603926643528750994292919, 2.45244224726155159119614850906, 3.50456003493790340081758617816, 4.20702906396356179995133393907, 4.79613870249856371703601441021, 6.08874624632761653353625403548, 6.63695554534858477095513856212, 7.41804175913356590854194182500, 8.087879855501762398758829369040
