| L(s) = 1 | + 3-s + 5-s − 0.239·7-s + 9-s + 4.45·11-s + 6.75·13-s + 15-s + 0.115·17-s + 0.902·19-s − 0.239·21-s + 8.04·23-s + 25-s + 27-s − 6.28·29-s + 5.30·31-s + 4.45·33-s − 0.239·35-s − 11.4·37-s + 6.75·39-s + 1.22·41-s − 3.53·43-s + 45-s − 4.40·47-s − 6.94·49-s + 0.115·51-s − 6.23·53-s + 4.45·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 0.0904·7-s + 0.333·9-s + 1.34·11-s + 1.87·13-s + 0.258·15-s + 0.0280·17-s + 0.206·19-s − 0.0522·21-s + 1.67·23-s + 0.200·25-s + 0.192·27-s − 1.16·29-s + 0.952·31-s + 0.776·33-s − 0.0404·35-s − 1.87·37-s + 1.08·39-s + 0.190·41-s − 0.538·43-s + 0.149·45-s − 0.643·47-s − 0.991·49-s + 0.0161·51-s − 0.855·53-s + 0.601·55-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
| \(L(1)\) |
\(\approx\) |
\(3.281809545\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.281809545\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.239T + 7T^{2} \) |
| 11 | \( 1 - 4.45T + 11T^{2} \) |
| 13 | \( 1 - 6.75T + 13T^{2} \) |
| 17 | \( 1 - 0.115T + 17T^{2} \) |
| 19 | \( 1 - 0.902T + 19T^{2} \) |
| 23 | \( 1 - 8.04T + 23T^{2} \) |
| 29 | \( 1 + 6.28T + 29T^{2} \) |
| 31 | \( 1 - 5.30T + 31T^{2} \) |
| 37 | \( 1 + 11.4T + 37T^{2} \) |
| 41 | \( 1 - 1.22T + 41T^{2} \) |
| 43 | \( 1 + 3.53T + 43T^{2} \) |
| 47 | \( 1 + 4.40T + 47T^{2} \) |
| 53 | \( 1 + 6.23T + 53T^{2} \) |
| 59 | \( 1 + 8.41T + 59T^{2} \) |
| 61 | \( 1 - 5.35T + 61T^{2} \) |
| 71 | \( 1 - 6.97T + 71T^{2} \) |
| 73 | \( 1 - 7.40T + 73T^{2} \) |
| 79 | \( 1 + 14.9T + 79T^{2} \) |
| 83 | \( 1 + 1.14T + 83T^{2} \) |
| 89 | \( 1 - 9.99T + 89T^{2} \) |
| 97 | \( 1 - 5.69T + 97T^{2} \) |
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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
−8.700053213203047667974280956593, −7.81548978653157679732118208018, −6.74808741171017007034333107872, −6.48028402183392568024485082779, −5.52287161453335792242547710198, −4.59833141290640017031976590662, −3.57036480508435668868599142864, −3.23014573216680997480781041233, −1.78172177507753736377305033964, −1.15224035206312284303481680784,
1.15224035206312284303481680784, 1.78172177507753736377305033964, 3.23014573216680997480781041233, 3.57036480508435668868599142864, 4.59833141290640017031976590662, 5.52287161453335792242547710198, 6.48028402183392568024485082779, 6.74808741171017007034333107872, 7.81548978653157679732118208018, 8.700053213203047667974280956593
