| L(s) = 1 | + 1.20·2-s − 0.545·4-s − 5-s − 4.31·7-s − 3.06·8-s − 1.20·10-s − 6.18·13-s − 5.20·14-s − 2.61·16-s − 0.803·17-s + 1.87·19-s + 0.545·20-s − 8.08·23-s + 25-s − 7.45·26-s + 2.35·28-s − 1.05·29-s − 1.61·31-s + 2.99·32-s − 0.969·34-s + 4.31·35-s − 5.54·37-s + 2.25·38-s + 3.06·40-s + 5.05·41-s + 9.87·43-s − 9.74·46-s + ⋯ |
| L(s) = 1 | + 0.852·2-s − 0.272·4-s − 0.447·5-s − 1.63·7-s − 1.08·8-s − 0.381·10-s − 1.71·13-s − 1.39·14-s − 0.652·16-s − 0.194·17-s + 0.429·19-s + 0.122·20-s − 1.68·23-s + 0.200·25-s − 1.46·26-s + 0.445·28-s − 0.196·29-s − 0.290·31-s + 0.528·32-s − 0.166·34-s + 0.729·35-s − 0.910·37-s + 0.366·38-s + 0.485·40-s + 0.788·41-s + 1.50·43-s − 1.43·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5158742591\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5158742591\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 1.20T + 2T^{2} \) |
| 7 | \( 1 + 4.31T + 7T^{2} \) |
| 13 | \( 1 + 6.18T + 13T^{2} \) |
| 17 | \( 1 + 0.803T + 17T^{2} \) |
| 19 | \( 1 - 1.87T + 19T^{2} \) |
| 23 | \( 1 + 8.08T + 23T^{2} \) |
| 29 | \( 1 + 1.05T + 29T^{2} \) |
| 31 | \( 1 + 1.61T + 31T^{2} \) |
| 37 | \( 1 + 5.54T + 37T^{2} \) |
| 41 | \( 1 - 5.05T + 41T^{2} \) |
| 43 | \( 1 - 9.87T + 43T^{2} \) |
| 47 | \( 1 + 12.9T + 47T^{2} \) |
| 53 | \( 1 - 2.66T + 53T^{2} \) |
| 59 | \( 1 + 6.60T + 59T^{2} \) |
| 61 | \( 1 - 5.34T + 61T^{2} \) |
| 67 | \( 1 + 7.49T + 67T^{2} \) |
| 71 | \( 1 - 5.82T + 71T^{2} \) |
| 73 | \( 1 + 4.80T + 73T^{2} \) |
| 79 | \( 1 - 13.9T + 79T^{2} \) |
| 83 | \( 1 - 12.0T + 83T^{2} \) |
| 89 | \( 1 + 12.7T + 89T^{2} \) |
| 97 | \( 1 - 1.34T + 97T^{2} \) |
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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
−8.046471973542444699574488323323, −7.32456046448461438552724462842, −6.59814228936070153012796854831, −5.93731305718501037696459556832, −5.22394428105086629433801993041, −4.40541266239947266564454470004, −3.74206249874256422615378579353, −3.06450935155589381887037595954, −2.29730606046530322848783065816, −0.31290428022100945947333596582,
0.31290428022100945947333596582, 2.29730606046530322848783065816, 3.06450935155589381887037595954, 3.74206249874256422615378579353, 4.40541266239947266564454470004, 5.22394428105086629433801993041, 5.93731305718501037696459556832, 6.59814228936070153012796854831, 7.32456046448461438552724462842, 8.046471973542444699574488323323
