L(s) = 1 | − 0.550·2-s − 2.20·3-s − 1.69·4-s − 3.11·5-s + 1.21·6-s − 3.99·7-s + 2.03·8-s + 1.86·9-s + 1.71·10-s + 5.48·11-s + 3.74·12-s + 13-s + 2.19·14-s + 6.86·15-s + 2.27·16-s − 0.435·17-s − 1.02·18-s + 0.374·19-s + 5.28·20-s + 8.80·21-s − 3.01·22-s + 5.48·23-s − 4.48·24-s + 4.70·25-s − 0.550·26-s + 2.51·27-s + 6.77·28-s + ⋯ |
L(s) = 1 | − 0.389·2-s − 1.27·3-s − 0.848·4-s − 1.39·5-s + 0.495·6-s − 1.50·7-s + 0.719·8-s + 0.620·9-s + 0.542·10-s + 1.65·11-s + 1.08·12-s + 0.277·13-s + 0.587·14-s + 1.77·15-s + 0.568·16-s − 0.105·17-s − 0.241·18-s + 0.0859·19-s + 1.18·20-s + 1.92·21-s − 0.643·22-s + 1.14·23-s − 0.915·24-s + 0.940·25-s − 0.107·26-s + 0.483·27-s + 1.28·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 13 | \( 1 - T \) |
| 103 | \( 1 - T \) |
good | 2 | \( 1 + 0.550T + 2T^{2} \) |
| 3 | \( 1 + 2.20T + 3T^{2} \) |
| 5 | \( 1 + 3.11T + 5T^{2} \) |
| 7 | \( 1 + 3.99T + 7T^{2} \) |
| 11 | \( 1 - 5.48T + 11T^{2} \) |
| 17 | \( 1 + 0.435T + 17T^{2} \) |
| 19 | \( 1 - 0.374T + 19T^{2} \) |
| 23 | \( 1 - 5.48T + 23T^{2} \) |
| 29 | \( 1 + 3.21T + 29T^{2} \) |
| 31 | \( 1 + 10.4T + 31T^{2} \) |
| 37 | \( 1 - 8.02T + 37T^{2} \) |
| 41 | \( 1 + 1.66T + 41T^{2} \) |
| 43 | \( 1 + 2.78T + 43T^{2} \) |
| 47 | \( 1 + 8.82T + 47T^{2} \) |
| 53 | \( 1 - 14.4T + 53T^{2} \) |
| 59 | \( 1 - 12.4T + 59T^{2} \) |
| 61 | \( 1 + 13.7T + 61T^{2} \) |
| 67 | \( 1 - 0.833T + 67T^{2} \) |
| 71 | \( 1 + 1.81T + 71T^{2} \) |
| 73 | \( 1 - 10.8T + 73T^{2} \) |
| 79 | \( 1 - 6.31T + 79T^{2} \) |
| 83 | \( 1 - 6.28T + 83T^{2} \) |
| 89 | \( 1 + 6.83T + 89T^{2} \) |
| 97 | \( 1 - 5.06T + 97T^{2} \) |
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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
−9.210896724973836973047050051692, −8.634573758913120411646305692724, −7.40511690291298152957372185884, −6.78198014328695664021697059744, −5.97643467325010901408176512031, −4.95032610025182787765444003226, −3.91347251310425072846800945304, −3.50755845879417323013201644914, −0.954158189066327071383561429542, 0,
0.954158189066327071383561429542, 3.50755845879417323013201644914, 3.91347251310425072846800945304, 4.95032610025182787765444003226, 5.97643467325010901408176512031, 6.78198014328695664021697059744, 7.40511690291298152957372185884, 8.634573758913120411646305692724, 9.210896724973836973047050051692