| L(s) = 1 | + 3·3-s − 1.46·5-s − 8.39·7-s + 9·9-s − 34.7·11-s + 13·13-s − 4.39·15-s − 108.·17-s − 143.·19-s − 25.1·21-s − 128.·23-s − 122.·25-s + 27·27-s + 18.8·29-s − 78.5·31-s − 104.·33-s + 12.2·35-s + 327.·37-s + 39·39-s + 327.·41-s + 336.·43-s − 13.1·45-s + 99.2·47-s − 272.·49-s − 324.·51-s + 686.·53-s + 50.9·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.130·5-s − 0.453·7-s + 0.333·9-s − 0.953·11-s + 0.277·13-s − 0.0756·15-s − 1.54·17-s − 1.72·19-s − 0.261·21-s − 1.16·23-s − 0.982·25-s + 0.192·27-s + 0.120·29-s − 0.455·31-s − 0.550·33-s + 0.0593·35-s + 1.45·37-s + 0.160·39-s + 1.24·41-s + 1.19·43-s − 0.0436·45-s + 0.308·47-s − 0.794·49-s − 0.890·51-s + 1.77·53-s + 0.124·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.428547025\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.428547025\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 3T \) |
| 13 | \( 1 - 13T \) |
| good | 5 | \( 1 + 1.46T + 125T^{2} \) |
| 7 | \( 1 + 8.39T + 343T^{2} \) |
| 11 | \( 1 + 34.7T + 1.33e3T^{2} \) |
| 17 | \( 1 + 108.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 143.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 128.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 18.8T + 2.43e4T^{2} \) |
| 31 | \( 1 + 78.5T + 2.97e4T^{2} \) |
| 37 | \( 1 - 327.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 327.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 336.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 99.2T + 1.03e5T^{2} \) |
| 53 | \( 1 - 686.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 242.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 644.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 871.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 100.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 604.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.07e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 741.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 501.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.56e3T + 9.12e5T^{2} \) |
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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
−8.384737634409435292227627961647, −8.063548152737621392501419961613, −7.04763235369882928778174309826, −6.31932603594109436567061867419, −5.54859452769672366399960705513, −4.21325825265287246956331878583, −3.98827373805609284950387952220, −2.47584528629301936891550709142, −2.20327455275982344188866013543, −0.47674032643355995355943430780,
0.47674032643355995355943430780, 2.20327455275982344188866013543, 2.47584528629301936891550709142, 3.98827373805609284950387952220, 4.21325825265287246956331878583, 5.54859452769672366399960705513, 6.31932603594109436567061867419, 7.04763235369882928778174309826, 8.063548152737621392501419961613, 8.384737634409435292227627961647
