| L(s) = 1 | − 3-s − 2·4-s − 5-s − 7-s + 9-s + 2·12-s + 15-s + 4·16-s − 3·17-s + 6·19-s + 2·20-s + 21-s − 8·23-s − 4·25-s − 27-s + 2·28-s − 29-s + 6·31-s + 35-s − 2·36-s − 2·37-s + 6·41-s + 43-s − 45-s + 3·47-s − 4·48-s + 49-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 4-s − 0.447·5-s − 0.377·7-s + 1/3·9-s + 0.577·12-s + 0.258·15-s + 16-s − 0.727·17-s + 1.37·19-s + 0.447·20-s + 0.218·21-s − 1.66·23-s − 4/5·25-s − 0.192·27-s + 0.377·28-s − 0.185·29-s + 1.07·31-s + 0.169·35-s − 1/3·36-s − 0.328·37-s + 0.937·41-s + 0.152·43-s − 0.149·45-s + 0.437·47-s − 0.577·48-s + 1/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 73689 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 73689 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9291621364\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9291621364\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 7 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 + 11 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 - T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−13.85103368913221, −13.59450100146687, −13.23449329371066, −12.45366427748377, −12.09360891279310, −11.71429940549840, −11.18697031719410, −10.42532977822026, −9.976778916008871, −9.683265957410972, −9.040285905196879, −8.480985501334303, −7.962628708445621, −7.439077458652366, −6.919742533100538, −6.069070618257440, −5.781916201835858, −5.168098849885346, −4.540130912086087, −3.922186222100912, −3.720782647232246, −2.759713122502855, −1.999613713141527, −0.9699057316691078, −0.4204272057605798,
0.4204272057605798, 0.9699057316691078, 1.999613713141527, 2.759713122502855, 3.720782647232246, 3.922186222100912, 4.540130912086087, 5.168098849885346, 5.781916201835858, 6.069070618257440, 6.919742533100538, 7.439077458652366, 7.962628708445621, 8.480985501334303, 9.040285905196879, 9.683265957410972, 9.976778916008871, 10.42532977822026, 11.18697031719410, 11.71429940549840, 12.09360891279310, 12.45366427748377, 13.23449329371066, 13.59450100146687, 13.85103368913221
