| L(s) = 1 | + 0.333·2-s + 1.03·3-s − 1.88·4-s + 0.343·6-s − 2.91·7-s − 1.29·8-s − 1.93·9-s − 5.85·11-s − 1.94·12-s − 5.25·13-s − 0.970·14-s + 3.34·16-s − 1.44·17-s − 0.645·18-s − 4.06·19-s − 3.00·21-s − 1.94·22-s − 1.01·23-s − 1.33·24-s − 1.75·26-s − 5.09·27-s + 5.50·28-s − 7.18·29-s + 6.93·31-s + 3.70·32-s − 6.03·33-s − 0.480·34-s + ⋯ |
| L(s) = 1 | + 0.235·2-s + 0.595·3-s − 0.944·4-s + 0.140·6-s − 1.10·7-s − 0.457·8-s − 0.645·9-s − 1.76·11-s − 0.562·12-s − 1.45·13-s − 0.259·14-s + 0.836·16-s − 0.350·17-s − 0.152·18-s − 0.932·19-s − 0.655·21-s − 0.415·22-s − 0.212·23-s − 0.272·24-s − 0.343·26-s − 0.979·27-s + 1.04·28-s − 1.33·29-s + 1.24·31-s + 0.655·32-s − 1.05·33-s − 0.0824·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1497915591\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1497915591\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 241 | \( 1 - T \) |
| good | 2 | \( 1 - 0.333T + 2T^{2} \) |
| 3 | \( 1 - 1.03T + 3T^{2} \) |
| 7 | \( 1 + 2.91T + 7T^{2} \) |
| 11 | \( 1 + 5.85T + 11T^{2} \) |
| 13 | \( 1 + 5.25T + 13T^{2} \) |
| 17 | \( 1 + 1.44T + 17T^{2} \) |
| 19 | \( 1 + 4.06T + 19T^{2} \) |
| 23 | \( 1 + 1.01T + 23T^{2} \) |
| 29 | \( 1 + 7.18T + 29T^{2} \) |
| 31 | \( 1 - 6.93T + 31T^{2} \) |
| 37 | \( 1 - 5.29T + 37T^{2} \) |
| 41 | \( 1 + 4.45T + 41T^{2} \) |
| 43 | \( 1 + 3.29T + 43T^{2} \) |
| 47 | \( 1 + 1.96T + 47T^{2} \) |
| 53 | \( 1 + 0.430T + 53T^{2} \) |
| 59 | \( 1 - 8.44T + 59T^{2} \) |
| 61 | \( 1 - 3.27T + 61T^{2} \) |
| 67 | \( 1 + 13.7T + 67T^{2} \) |
| 71 | \( 1 - 4.02T + 71T^{2} \) |
| 73 | \( 1 - 3.64T + 73T^{2} \) |
| 79 | \( 1 + 8.06T + 79T^{2} \) |
| 83 | \( 1 - 9.94T + 83T^{2} \) |
| 89 | \( 1 + 7.05T + 89T^{2} \) |
| 97 | \( 1 - 3.07T + 97T^{2} \) |
| 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
−8.093978849500382668344953761478, −7.60825704200517134724082079552, −6.61788297104664857641936820107, −5.78510839214622643991228048759, −5.16826901591690253454705801919, −4.47307325505020149482541889444, −3.53865355460767222210427908813, −2.78896664905849571949889700100, −2.30800384580157912956465349851, −0.17324049693276903609682482946,
0.17324049693276903609682482946, 2.30800384580157912956465349851, 2.78896664905849571949889700100, 3.53865355460767222210427908813, 4.47307325505020149482541889444, 5.16826901591690253454705801919, 5.78510839214622643991228048759, 6.61788297104664857641936820107, 7.60825704200517134724082079552, 8.093978849500382668344953761478
