| L(s) = 1 | − 3·3-s − 4.10·5-s − 28.9·7-s + 9·9-s + 20.5·11-s − 64.4·13-s + 12.3·15-s − 41.9·17-s + 19·19-s + 86.7·21-s + 82.4·23-s − 108.·25-s − 27·27-s + 23.6·29-s + 200.·31-s − 61.5·33-s + 118.·35-s − 189.·37-s + 193.·39-s + 438.·41-s + 406.·43-s − 36.9·45-s − 272.·47-s + 493.·49-s + 125.·51-s − 183.·53-s − 84.1·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.367·5-s − 1.56·7-s + 0.333·9-s + 0.562·11-s − 1.37·13-s + 0.211·15-s − 0.597·17-s + 0.229·19-s + 0.901·21-s + 0.747·23-s − 0.865·25-s − 0.192·27-s + 0.151·29-s + 1.16·31-s − 0.324·33-s + 0.573·35-s − 0.840·37-s + 0.793·39-s + 1.67·41-s + 1.44·43-s − 0.122·45-s − 0.845·47-s + 1.43·49-s + 0.345·51-s − 0.476·53-s − 0.206·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.8013035647\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8013035647\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| 19 | \( 1 - 19T \) |
| good | 5 | \( 1 + 4.10T + 125T^{2} \) |
| 7 | \( 1 + 28.9T + 343T^{2} \) |
| 11 | \( 1 - 20.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + 64.4T + 2.19e3T^{2} \) |
| 17 | \( 1 + 41.9T + 4.91e3T^{2} \) |
| 23 | \( 1 - 82.4T + 1.21e4T^{2} \) |
| 29 | \( 1 - 23.6T + 2.43e4T^{2} \) |
| 31 | \( 1 - 200.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 189.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 438.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 406.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 272.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 183.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 57.0T + 2.05e5T^{2} \) |
| 61 | \( 1 - 304.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 523.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 527.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 104.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 837.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.29e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.20e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 626.T + 9.12e5T^{2} \) |
| 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
−10.61416281262908471672473527913, −9.697818945830367701482711537409, −9.177836555128775955404209342067, −7.69381929371229379282025852675, −6.82127173528814191846540155993, −6.12976989297491241427315245431, −4.87565234948530334402602056998, −3.78030753124890285977706663961, −2.56273103515264356670476290436, −0.55883155087367316270460399752,
0.55883155087367316270460399752, 2.56273103515264356670476290436, 3.78030753124890285977706663961, 4.87565234948530334402602056998, 6.12976989297491241427315245431, 6.82127173528814191846540155993, 7.69381929371229379282025852675, 9.177836555128775955404209342067, 9.697818945830367701482711537409, 10.61416281262908471672473527913
