| L(s) = 1 | − 14.6·3-s + 72.1·5-s − 27.4·9-s − 246.·11-s + 1.13e3·13-s − 1.05e3·15-s − 1.64e3·17-s − 1.64e3·19-s − 469.·23-s + 2.08e3·25-s + 3.97e3·27-s + 5.81e3·29-s − 5.73e3·31-s + 3.61e3·33-s + 435.·37-s − 1.66e4·39-s − 1.12e4·41-s + 6.14e3·43-s − 1.98e3·45-s + 1.46e4·47-s + 2.41e4·51-s + 2.68e4·53-s − 1.77e4·55-s + 2.41e4·57-s − 2.50e4·59-s − 1.57e4·61-s + 8.16e4·65-s + ⋯ |
| L(s) = 1 | − 0.941·3-s + 1.29·5-s − 0.112·9-s − 0.613·11-s + 1.85·13-s − 1.21·15-s − 1.37·17-s − 1.04·19-s − 0.184·23-s + 0.666·25-s + 1.04·27-s + 1.28·29-s − 1.07·31-s + 0.578·33-s + 0.0522·37-s − 1.74·39-s − 1.04·41-s + 0.507·43-s − 0.145·45-s + 0.965·47-s + 1.29·51-s + 1.31·53-s − 0.792·55-s + 0.985·57-s − 0.937·59-s − 0.541·61-s + 2.39·65-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
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + 14.6T + 243T^{2} \) |
| 5 | \( 1 - 72.1T + 3.12e3T^{2} \) |
| 11 | \( 1 + 246.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 1.13e3T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.64e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.64e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 469.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 5.81e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 5.73e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 435.T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.12e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 6.14e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.46e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.68e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.50e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.57e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 2.02e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 6.95e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.04e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 9.04e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 3.08e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.64e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.03e5T + 8.58e9T^{2} \) |
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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
−10.47475033261529624731123684346, −9.028995435026990090771893721826, −8.456130282123192153075262108845, −6.74530861250653223330930017911, −6.09330184313754388334766122942, −5.47419455080797805579511735418, −4.24975838161690584905367343796, −2.60510593734887236196278127738, −1.42555878854647274772050253411, 0,
1.42555878854647274772050253411, 2.60510593734887236196278127738, 4.24975838161690584905367343796, 5.47419455080797805579511735418, 6.09330184313754388334766122942, 6.74530861250653223330930017911, 8.456130282123192153075262108845, 9.028995435026990090771893721826, 10.47475033261529624731123684346
