L(s) = 1 | − 2.75·2-s − 0.459·3-s + 5.56·4-s + 5-s + 1.26·6-s + 0.587·7-s − 9.81·8-s − 2.78·9-s − 2.75·10-s − 0.892·11-s − 2.56·12-s − 1.31·13-s − 1.61·14-s − 0.459·15-s + 15.8·16-s − 6.89·17-s + 7.67·18-s + 2.78·19-s + 5.56·20-s − 0.270·21-s + 2.45·22-s − 0.636·23-s + 4.51·24-s + 25-s + 3.61·26-s + 2.66·27-s + 3.27·28-s + ⋯ |
L(s) = 1 | − 1.94·2-s − 0.265·3-s + 2.78·4-s + 0.447·5-s + 0.516·6-s + 0.222·7-s − 3.47·8-s − 0.929·9-s − 0.869·10-s − 0.268·11-s − 0.739·12-s − 0.364·13-s − 0.432·14-s − 0.118·15-s + 3.96·16-s − 1.67·17-s + 1.80·18-s + 0.639·19-s + 1.24·20-s − 0.0589·21-s + 0.523·22-s − 0.132·23-s + 0.921·24-s + 0.200·25-s + 0.709·26-s + 0.512·27-s + 0.618·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 445 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 445 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 89 | \( 1 - T \) |
good | 2 | \( 1 + 2.75T + 2T^{2} \) |
| 3 | \( 1 + 0.459T + 3T^{2} \) |
| 7 | \( 1 - 0.587T + 7T^{2} \) |
| 11 | \( 1 + 0.892T + 11T^{2} \) |
| 13 | \( 1 + 1.31T + 13T^{2} \) |
| 17 | \( 1 + 6.89T + 17T^{2} \) |
| 19 | \( 1 - 2.78T + 19T^{2} \) |
| 23 | \( 1 + 0.636T + 23T^{2} \) |
| 29 | \( 1 - 5.55T + 29T^{2} \) |
| 31 | \( 1 + 9.03T + 31T^{2} \) |
| 37 | \( 1 - 0.632T + 37T^{2} \) |
| 41 | \( 1 + 7.36T + 41T^{2} \) |
| 43 | \( 1 + 7.41T + 43T^{2} \) |
| 47 | \( 1 + 6.16T + 47T^{2} \) |
| 53 | \( 1 - 3.18T + 53T^{2} \) |
| 59 | \( 1 - 5.68T + 59T^{2} \) |
| 61 | \( 1 + 4.98T + 61T^{2} \) |
| 67 | \( 1 - 7.78T + 67T^{2} \) |
| 71 | \( 1 + 14.9T + 71T^{2} \) |
| 73 | \( 1 - 10.9T + 73T^{2} \) |
| 79 | \( 1 + 11.7T + 79T^{2} \) |
| 83 | \( 1 + 14.3T + 83T^{2} \) |
| 97 | \( 1 - 5.86T + 97T^{2} \) |
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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
−10.50135114093515880110584856330, −9.692542928512956046648912166420, −8.815249615773039709526197231051, −8.256213937658395535298854455860, −7.10421504866325048064708776598, −6.36944103482320413140153797868, −5.25319731601321216044861769518, −2.93780312850244441451367290352, −1.83296779380478863816478260892, 0,
1.83296779380478863816478260892, 2.93780312850244441451367290352, 5.25319731601321216044861769518, 6.36944103482320413140153797868, 7.10421504866325048064708776598, 8.256213937658395535298854455860, 8.815249615773039709526197231051, 9.692542928512956046648912166420, 10.50135114093515880110584856330