| L(s) = 1 | − 3-s − 2.79·5-s − 7-s + 9-s − 3.78·11-s − 2.95·13-s + 2.79·15-s − 3.46·17-s − 19-s + 21-s + 3.78·23-s + 2.80·25-s − 27-s − 6.57·29-s − 5.78·31-s + 3.78·33-s + 2.79·35-s − 9.84·37-s + 2.95·39-s + 1.30·41-s − 10.0·43-s − 2.79·45-s + 6.99·47-s + 49-s + 3.46·51-s − 4.63·53-s + 10.5·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.24·5-s − 0.377·7-s + 0.333·9-s − 1.14·11-s − 0.818·13-s + 0.721·15-s − 0.840·17-s − 0.229·19-s + 0.218·21-s + 0.789·23-s + 0.560·25-s − 0.192·27-s − 1.22·29-s − 1.03·31-s + 0.658·33-s + 0.472·35-s − 1.61·37-s + 0.472·39-s + 0.203·41-s − 1.53·43-s − 0.416·45-s + 1.01·47-s + 0.142·49-s + 0.484·51-s − 0.637·53-s + 1.42·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.05829160733\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.05829160733\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 + 2.79T + 5T^{2} \) |
| 11 | \( 1 + 3.78T + 11T^{2} \) |
| 13 | \( 1 + 2.95T + 13T^{2} \) |
| 17 | \( 1 + 3.46T + 17T^{2} \) |
| 23 | \( 1 - 3.78T + 23T^{2} \) |
| 29 | \( 1 + 6.57T + 29T^{2} \) |
| 31 | \( 1 + 5.78T + 31T^{2} \) |
| 37 | \( 1 + 9.84T + 37T^{2} \) |
| 41 | \( 1 - 1.30T + 41T^{2} \) |
| 43 | \( 1 + 10.0T + 43T^{2} \) |
| 47 | \( 1 - 6.99T + 47T^{2} \) |
| 53 | \( 1 + 4.63T + 53T^{2} \) |
| 59 | \( 1 + 7.57T + 59T^{2} \) |
| 61 | \( 1 - 2.67T + 61T^{2} \) |
| 67 | \( 1 + 10.0T + 67T^{2} \) |
| 71 | \( 1 + 10.2T + 71T^{2} \) |
| 73 | \( 1 - 5.64T + 73T^{2} \) |
| 79 | \( 1 + 16.3T + 79T^{2} \) |
| 83 | \( 1 - 8.65T + 83T^{2} \) |
| 89 | \( 1 - 12.4T + 89T^{2} \) |
| 97 | \( 1 + 16.6T + 97T^{2} \) |
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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
−7.85172098367119332982855322523, −7.25880197880317521156687849825, −6.88055921911620563803847256478, −5.79127203675415046678387974615, −5.11853316449337015045425248677, −4.48930476882124282803637710296, −3.66079010128630687790906645005, −2.88997114071676526237611020342, −1.81173588924814516266101391912, −0.12205572937481378192866232192,
0.12205572937481378192866232192, 1.81173588924814516266101391912, 2.88997114071676526237611020342, 3.66079010128630687790906645005, 4.48930476882124282803637710296, 5.11853316449337015045425248677, 5.79127203675415046678387974615, 6.88055921911620563803847256478, 7.25880197880317521156687849825, 7.85172098367119332982855322523
