L(s) = 1 | − 16.3·3-s − 51.0·5-s + 24.8·9-s − 273.·11-s + 369.·13-s + 834.·15-s + 1.03e3·17-s + 2.18e3·19-s − 1.01e3·23-s − 522.·25-s + 3.57e3·27-s − 212.·29-s − 229.·31-s + 4.47e3·33-s − 9.11e3·37-s − 6.04e3·39-s + 1.67e4·41-s + 1.61e4·43-s − 1.26e3·45-s − 1.12e3·47-s − 1.69e4·51-s + 3.41e4·53-s + 1.39e4·55-s − 3.57e4·57-s − 1.11e4·59-s − 3.59e4·61-s − 1.88e4·65-s + ⋯ |
L(s) = 1 | − 1.04·3-s − 0.912·5-s + 0.102·9-s − 0.681·11-s + 0.605·13-s + 0.958·15-s + 0.869·17-s + 1.38·19-s − 0.399·23-s − 0.167·25-s + 0.942·27-s − 0.0469·29-s − 0.0429·31-s + 0.715·33-s − 1.09·37-s − 0.636·39-s + 1.55·41-s + 1.32·43-s − 0.0933·45-s − 0.0743·47-s − 0.913·51-s + 1.67·53-s + 0.621·55-s − 1.45·57-s − 0.418·59-s − 1.23·61-s − 0.552·65-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 | \( 1 + 16.3T + 243T^{2} \) |
| 5 | \( 1 + 51.0T + 3.12e3T^{2} \) |
| 11 | \( 1 + 273.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 369.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.03e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.18e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 1.01e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 212.T + 2.05e7T^{2} \) |
| 31 | \( 1 + 229.T + 2.86e7T^{2} \) |
| 37 | \( 1 + 9.11e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.67e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.61e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.12e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.41e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.11e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.59e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 5.75e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 2.09e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 5.64e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 4.81e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 6.43e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 6.24e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 3.52e4T + 8.58e9T^{2} \) |
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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
−10.35375095540583397240512892756, −9.130187807150539508891701900149, −7.946048043752672763088154544129, −7.31636115519599865669556792819, −5.95986427076537419642263024327, −5.34416401372581503624317134121, −4.12302948938134592315891305340, −2.98130972564644835174711979350, −1.06861629131071351347065746094, 0,
1.06861629131071351347065746094, 2.98130972564644835174711979350, 4.12302948938134592315891305340, 5.34416401372581503624317134121, 5.95986427076537419642263024327, 7.31636115519599865669556792819, 7.946048043752672763088154544129, 9.130187807150539508891701900149, 10.35375095540583397240512892756