| L(s) = 1 | − 19.3·3-s + 25·5-s − 49·7-s + 129.·9-s + 10.9·11-s + 29.6·13-s − 482.·15-s − 432.·17-s + 956.·19-s + 945.·21-s − 979.·23-s + 625·25-s + 2.19e3·27-s + 996.·29-s − 4.79e3·31-s − 210.·33-s − 1.22e3·35-s − 1.88e3·37-s − 573.·39-s − 1.92e3·41-s + 1.80e4·43-s + 3.23e3·45-s + 2.85e4·47-s + 2.40e3·49-s + 8.34e3·51-s − 287.·53-s + 272.·55-s + ⋯ |
| L(s) = 1 | − 1.23·3-s + 0.447·5-s − 0.377·7-s + 0.532·9-s + 0.0271·11-s + 0.0487·13-s − 0.553·15-s − 0.362·17-s + 0.607·19-s + 0.467·21-s − 0.386·23-s + 0.200·25-s + 0.578·27-s + 0.220·29-s − 0.895·31-s − 0.0336·33-s − 0.169·35-s − 0.226·37-s − 0.0603·39-s − 0.179·41-s + 1.49·43-s + 0.238·45-s + 1.88·47-s + 0.142·49-s + 0.449·51-s − 0.0140·53-s + 0.0121·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - 25T \) |
| 7 | \( 1 + 49T \) |
| good | 3 | \( 1 + 19.3T + 243T^{2} \) |
| 11 | \( 1 - 10.9T + 1.61e5T^{2} \) |
| 13 | \( 1 - 29.6T + 3.71e5T^{2} \) |
| 17 | \( 1 + 432.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 956.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 979.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 996.T + 2.05e7T^{2} \) |
| 31 | \( 1 + 4.79e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.88e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.92e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.80e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.85e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 287.T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.12e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.28e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.70e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 6.39e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.91e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 7.12e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 9.43e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 7.86e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 9.34e4T + 8.58e9T^{2} \) |
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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
−9.662863099818797725432573087190, −8.837129944107051632967414787126, −7.52593870498822479765060161590, −6.59203842449609745957997425523, −5.83045269150947931037442139669, −5.13716378062526209022329426172, −3.94688784427992563561491519719, −2.53705547032818661024207923637, −1.10968522333673518096152742048, 0,
1.10968522333673518096152742048, 2.53705547032818661024207923637, 3.94688784427992563561491519719, 5.13716378062526209022329426172, 5.83045269150947931037442139669, 6.59203842449609745957997425523, 7.52593870498822479765060161590, 8.837129944107051632967414787126, 9.662863099818797725432573087190
