| L(s) = 1 | + 3-s + 2.56·5-s − 3·7-s + 9-s − 2.56·11-s + 2·13-s + 2.56·15-s + 0.561·17-s − 19-s − 3·21-s + 2.56·23-s + 1.56·25-s + 27-s − 8.24·29-s − 5.12·31-s − 2.56·33-s − 7.68·35-s − 6.24·37-s + 2·39-s − 7.68·41-s − 11.1·43-s + 2.56·45-s − 7.56·47-s + 2·49-s + 0.561·51-s + 1.43·53-s − 6.56·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.14·5-s − 1.13·7-s + 0.333·9-s − 0.772·11-s + 0.554·13-s + 0.661·15-s + 0.136·17-s − 0.229·19-s − 0.654·21-s + 0.534·23-s + 0.312·25-s + 0.192·27-s − 1.53·29-s − 0.920·31-s − 0.445·33-s − 1.29·35-s − 1.02·37-s + 0.320·39-s − 1.20·41-s − 1.69·43-s + 0.381·45-s − 1.10·47-s + 0.285·49-s + 0.0786·51-s + 0.197·53-s − 0.884·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6096 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6096 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 127 | \( 1 + T \) |
| good | 5 | \( 1 - 2.56T + 5T^{2} \) |
| 7 | \( 1 + 3T + 7T^{2} \) |
| 11 | \( 1 + 2.56T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 - 0.561T + 17T^{2} \) |
| 19 | \( 1 + T + 19T^{2} \) |
| 23 | \( 1 - 2.56T + 23T^{2} \) |
| 29 | \( 1 + 8.24T + 29T^{2} \) |
| 31 | \( 1 + 5.12T + 31T^{2} \) |
| 37 | \( 1 + 6.24T + 37T^{2} \) |
| 41 | \( 1 + 7.68T + 41T^{2} \) |
| 43 | \( 1 + 11.1T + 43T^{2} \) |
| 47 | \( 1 + 7.56T + 47T^{2} \) |
| 53 | \( 1 - 1.43T + 53T^{2} \) |
| 59 | \( 1 - 4.68T + 59T^{2} \) |
| 61 | \( 1 - 9.12T + 61T^{2} \) |
| 67 | \( 1 + 6T + 67T^{2} \) |
| 71 | \( 1 + 2.43T + 71T^{2} \) |
| 73 | \( 1 - 10.6T + 73T^{2} \) |
| 79 | \( 1 - 11.3T + 79T^{2} \) |
| 83 | \( 1 + 2.43T + 83T^{2} \) |
| 89 | \( 1 - 5.80T + 89T^{2} \) |
| 97 | \( 1 + 4T + 97T^{2} \) |
| 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
−7.72911634677825573673525471117, −6.85754214454738208896394444794, −6.41840610022797099199409096989, −5.51519771917666612632321533990, −5.07477260761059618817959713005, −3.66743134563647689170105597856, −3.30116929734946766400710756419, −2.27589534199873250784120207123, −1.60893401179713458351172455905, 0,
1.60893401179713458351172455905, 2.27589534199873250784120207123, 3.30116929734946766400710756419, 3.66743134563647689170105597856, 5.07477260761059618817959713005, 5.51519771917666612632321533990, 6.41840610022797099199409096989, 6.85754214454738208896394444794, 7.72911634677825573673525471117
