L(s) = 1 | − 2-s − 3-s + 4-s + 2.32·5-s + 6-s − 0.928·7-s − 8-s + 9-s − 2.32·10-s + 6.48·11-s − 12-s − 4.80·13-s + 0.928·14-s − 2.32·15-s + 16-s − 0.708·17-s − 18-s + 19-s + 2.32·20-s + 0.928·21-s − 6.48·22-s + 3.31·23-s + 24-s + 0.384·25-s + 4.80·26-s − 27-s − 0.928·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.03·5-s + 0.408·6-s − 0.351·7-s − 0.353·8-s + 0.333·9-s − 0.733·10-s + 1.95·11-s − 0.288·12-s − 1.33·13-s + 0.248·14-s − 0.599·15-s + 0.250·16-s − 0.171·17-s − 0.235·18-s + 0.229·19-s + 0.518·20-s + 0.202·21-s − 1.38·22-s + 0.690·23-s + 0.204·24-s + 0.0768·25-s + 0.941·26-s − 0.192·27-s − 0.175·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| 53 | \( 1 - T \) |
good | 5 | \( 1 - 2.32T + 5T^{2} \) |
| 7 | \( 1 + 0.928T + 7T^{2} \) |
| 11 | \( 1 - 6.48T + 11T^{2} \) |
| 13 | \( 1 + 4.80T + 13T^{2} \) |
| 17 | \( 1 + 0.708T + 17T^{2} \) |
| 23 | \( 1 - 3.31T + 23T^{2} \) |
| 29 | \( 1 - 0.356T + 29T^{2} \) |
| 31 | \( 1 + 7.07T + 31T^{2} \) |
| 37 | \( 1 + 10.2T + 37T^{2} \) |
| 41 | \( 1 + 6.64T + 41T^{2} \) |
| 43 | \( 1 + 2.77T + 43T^{2} \) |
| 47 | \( 1 - 3.23T + 47T^{2} \) |
| 59 | \( 1 + 2.56T + 59T^{2} \) |
| 61 | \( 1 - 0.142T + 61T^{2} \) |
| 67 | \( 1 + 7.24T + 67T^{2} \) |
| 71 | \( 1 - 7.50T + 71T^{2} \) |
| 73 | \( 1 - 0.749T + 73T^{2} \) |
| 79 | \( 1 - 5.37T + 79T^{2} \) |
| 83 | \( 1 + 14.8T + 83T^{2} \) |
| 89 | \( 1 + 5.61T + 89T^{2} \) |
| 97 | \( 1 + 16.5T + 97T^{2} \) |
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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
−7.54323815407831336901463848624, −6.75636040979927622054693869702, −6.64608207912720757747871348632, −5.64689928656233672258388033288, −5.08585659968923685492026391136, −4.02429777065605835562059954370, −3.09837500123150586374012742075, −1.93969769661782771733266286315, −1.38019595959850813839538452364, 0,
1.38019595959850813839538452364, 1.93969769661782771733266286315, 3.09837500123150586374012742075, 4.02429777065605835562059954370, 5.08585659968923685492026391136, 5.64689928656233672258388033288, 6.64608207912720757747871348632, 6.75636040979927622054693869702, 7.54323815407831336901463848624