| L(s) = 1 | − 2.99·3-s + 103.·5-s − 234.·9-s + 233.·11-s + 204.·13-s − 308.·15-s + 112.·17-s + 2.48e3·19-s − 3.34e3·23-s + 7.48e3·25-s + 1.42e3·27-s − 4.55e3·29-s + 8.97e3·31-s − 698.·33-s + 3.78e3·37-s − 610.·39-s − 3.65e3·41-s + 2.11e4·43-s − 2.41e4·45-s − 7.79e3·47-s − 335.·51-s − 9.87e3·53-s + 2.40e4·55-s − 7.44e3·57-s − 2.07e4·59-s − 4.44e4·61-s + 2.10e4·65-s + ⋯ |
| L(s) = 1 | − 0.192·3-s + 1.84·5-s − 0.963·9-s + 0.581·11-s + 0.334·13-s − 0.353·15-s + 0.0940·17-s + 1.58·19-s − 1.31·23-s + 2.39·25-s + 0.376·27-s − 1.00·29-s + 1.67·31-s − 0.111·33-s + 0.454·37-s − 0.0643·39-s − 0.339·41-s + 1.74·43-s − 1.77·45-s − 0.514·47-s − 0.0180·51-s − 0.482·53-s + 1.07·55-s − 0.303·57-s − 0.775·59-s − 1.52·61-s + 0.617·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.967405875\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.967405875\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 2.99T + 243T^{2} \) |
| 5 | \( 1 - 103.T + 3.12e3T^{2} \) |
| 11 | \( 1 - 233.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 204.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 112.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.48e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 3.34e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 4.55e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 8.97e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 3.78e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 3.65e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 2.11e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 7.79e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 9.87e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.07e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 4.44e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.09e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 5.73e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 3.50e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 2.70e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 7.13e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 6.37e3T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.70e5T + 8.58e9T^{2} \) |
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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
−10.33953015212053350117581085643, −9.559159897654293762230252351856, −8.951058292650242652398203765217, −7.70289837682480659969767025362, −6.20880026099740968964871114131, −5.95947145482502370716880142994, −4.89060195506455352814222856056, −3.21883628150035374668142514754, −2.10271925653166264300869323697, −0.968343979785004941996778114524,
0.968343979785004941996778114524, 2.10271925653166264300869323697, 3.21883628150035374668142514754, 4.89060195506455352814222856056, 5.95947145482502370716880142994, 6.20880026099740968964871114131, 7.70289837682480659969767025362, 8.951058292650242652398203765217, 9.559159897654293762230252351856, 10.33953015212053350117581085643
