| L(s) = 1 | + 2.17·3-s + 5-s + 5.14·7-s + 1.73·9-s − 0.239·11-s − 2.92·13-s + 2.17·15-s + 3.35·17-s − 1.38·19-s + 11.2·21-s + 5.09·23-s + 25-s − 2.74·27-s − 7.74·29-s + 4.81·31-s − 0.521·33-s + 5.14·35-s + 1.45·37-s − 6.36·39-s − 6.02·41-s + 1.28·43-s + 1.73·45-s − 2.64·47-s + 19.4·49-s + 7.29·51-s + 10.7·53-s − 0.239·55-s + ⋯ |
| L(s) = 1 | + 1.25·3-s + 0.447·5-s + 1.94·7-s + 0.578·9-s − 0.0722·11-s − 0.810·13-s + 0.561·15-s + 0.813·17-s − 0.316·19-s + 2.44·21-s + 1.06·23-s + 0.200·25-s − 0.529·27-s − 1.43·29-s + 0.864·31-s − 0.0908·33-s + 0.870·35-s + 0.239·37-s − 1.01·39-s − 0.941·41-s + 0.195·43-s + 0.258·45-s − 0.386·47-s + 2.78·49-s + 1.02·51-s + 1.47·53-s − 0.0323·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.732771639\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.732771639\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 401 | \( 1 - T \) |
| good | 3 | \( 1 - 2.17T + 3T^{2} \) |
| 7 | \( 1 - 5.14T + 7T^{2} \) |
| 11 | \( 1 + 0.239T + 11T^{2} \) |
| 13 | \( 1 + 2.92T + 13T^{2} \) |
| 17 | \( 1 - 3.35T + 17T^{2} \) |
| 19 | \( 1 + 1.38T + 19T^{2} \) |
| 23 | \( 1 - 5.09T + 23T^{2} \) |
| 29 | \( 1 + 7.74T + 29T^{2} \) |
| 31 | \( 1 - 4.81T + 31T^{2} \) |
| 37 | \( 1 - 1.45T + 37T^{2} \) |
| 41 | \( 1 + 6.02T + 41T^{2} \) |
| 43 | \( 1 - 1.28T + 43T^{2} \) |
| 47 | \( 1 + 2.64T + 47T^{2} \) |
| 53 | \( 1 - 10.7T + 53T^{2} \) |
| 59 | \( 1 - 2.43T + 59T^{2} \) |
| 61 | \( 1 + 7.46T + 61T^{2} \) |
| 67 | \( 1 - 3.00T + 67T^{2} \) |
| 71 | \( 1 - 15.7T + 71T^{2} \) |
| 73 | \( 1 - 10.5T + 73T^{2} \) |
| 79 | \( 1 - 14.3T + 79T^{2} \) |
| 83 | \( 1 - 5.33T + 83T^{2} \) |
| 89 | \( 1 - 16.2T + 89T^{2} \) |
| 97 | \( 1 + 9.93T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.989472174335360913517163943096, −7.45069284216304206060515966557, −6.64816389717340019807249153717, −5.34676907645773105676375822632, −5.19287455757498783229957367626, −4.24219154893614242833299318632, −3.44084354277825838419820171615, −2.44068695344010665647590498992, −2.01760910470916319962025645042, −1.08292063376263097405531305899,
1.08292063376263097405531305899, 2.01760910470916319962025645042, 2.44068695344010665647590498992, 3.44084354277825838419820171615, 4.24219154893614242833299318632, 5.19287455757498783229957367626, 5.34676907645773105676375822632, 6.64816389717340019807249153717, 7.45069284216304206060515966557, 7.989472174335360913517163943096
