L(s) = 1 | − 2.06·2-s + 2.25·4-s − 3.73·5-s + 1.84·7-s − 0.536·8-s + 7.70·10-s + 4.36·11-s − 5.39·13-s − 3.80·14-s − 3.41·16-s − 4.13·17-s + 0.483·19-s − 8.43·20-s − 9.01·22-s + 2.47·23-s + 8.93·25-s + 11.1·26-s + 4.16·28-s + 8.03·29-s + 0.894·31-s + 8.11·32-s + 8.53·34-s − 6.88·35-s + 7.79·37-s − 0.997·38-s + 2.00·40-s − 1.77·41-s + ⋯ |
L(s) = 1 | − 1.45·2-s + 1.12·4-s − 1.66·5-s + 0.696·7-s − 0.189·8-s + 2.43·10-s + 1.31·11-s − 1.49·13-s − 1.01·14-s − 0.853·16-s − 1.00·17-s + 0.110·19-s − 1.88·20-s − 1.92·22-s + 0.515·23-s + 1.78·25-s + 2.18·26-s + 0.787·28-s + 1.49·29-s + 0.160·31-s + 1.43·32-s + 1.46·34-s − 1.16·35-s + 1.28·37-s − 0.161·38-s + 0.316·40-s − 0.277·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1143 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 127 | \( 1 - T \) |
good | 2 | \( 1 + 2.06T + 2T^{2} \) |
| 5 | \( 1 + 3.73T + 5T^{2} \) |
| 7 | \( 1 - 1.84T + 7T^{2} \) |
| 11 | \( 1 - 4.36T + 11T^{2} \) |
| 13 | \( 1 + 5.39T + 13T^{2} \) |
| 17 | \( 1 + 4.13T + 17T^{2} \) |
| 19 | \( 1 - 0.483T + 19T^{2} \) |
| 23 | \( 1 - 2.47T + 23T^{2} \) |
| 29 | \( 1 - 8.03T + 29T^{2} \) |
| 31 | \( 1 - 0.894T + 31T^{2} \) |
| 37 | \( 1 - 7.79T + 37T^{2} \) |
| 41 | \( 1 + 1.77T + 41T^{2} \) |
| 43 | \( 1 + 1.03T + 43T^{2} \) |
| 47 | \( 1 + 8.87T + 47T^{2} \) |
| 53 | \( 1 + 4.17T + 53T^{2} \) |
| 59 | \( 1 + 0.500T + 59T^{2} \) |
| 61 | \( 1 - 7.90T + 61T^{2} \) |
| 67 | \( 1 + 8.39T + 67T^{2} \) |
| 71 | \( 1 + 7.26T + 71T^{2} \) |
| 73 | \( 1 + 5.37T + 73T^{2} \) |
| 79 | \( 1 + 10.9T + 79T^{2} \) |
| 83 | \( 1 + 15.8T + 83T^{2} \) |
| 89 | \( 1 - 0.247T + 89T^{2} \) |
| 97 | \( 1 + 5.42T + 97T^{2} \) |
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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
−9.251354134319029355503198794862, −8.486634510222635608051571806390, −7.978694448020179450539183622301, −7.16958053400496203862024648827, −6.65904652323652706694366205904, −4.70343733553419295692001625380, −4.31682396594333930306249077955, −2.80310813857017585226562085145, −1.31988267610331545396315701572, 0,
1.31988267610331545396315701572, 2.80310813857017585226562085145, 4.31682396594333930306249077955, 4.70343733553419295692001625380, 6.65904652323652706694366205904, 7.16958053400496203862024648827, 7.978694448020179450539183622301, 8.486634510222635608051571806390, 9.251354134319029355503198794862