L(s) = 1 | − 2-s + 4-s + 3.58·5-s − 4.00·7-s − 8-s − 3.58·10-s − 1.14·11-s + 0.738·13-s + 4.00·14-s + 16-s − 4.19·17-s + 1.58·19-s + 3.58·20-s + 1.14·22-s + 2.22·23-s + 7.84·25-s − 0.738·26-s − 4.00·28-s − 2.40·29-s + 4.76·31-s − 32-s + 4.19·34-s − 14.3·35-s − 9.04·37-s − 1.58·38-s − 3.58·40-s − 3.19·41-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.5·4-s + 1.60·5-s − 1.51·7-s − 0.353·8-s − 1.13·10-s − 0.346·11-s + 0.204·13-s + 1.06·14-s + 0.250·16-s − 1.01·17-s + 0.364·19-s + 0.801·20-s + 0.244·22-s + 0.463·23-s + 1.56·25-s − 0.144·26-s − 0.756·28-s − 0.446·29-s + 0.855·31-s − 0.176·32-s + 0.719·34-s − 2.42·35-s − 1.48·37-s − 0.257·38-s − 0.566·40-s − 0.498·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 149 | \( 1 + T \) |
good | 5 | \( 1 - 3.58T + 5T^{2} \) |
| 7 | \( 1 + 4.00T + 7T^{2} \) |
| 11 | \( 1 + 1.14T + 11T^{2} \) |
| 13 | \( 1 - 0.738T + 13T^{2} \) |
| 17 | \( 1 + 4.19T + 17T^{2} \) |
| 19 | \( 1 - 1.58T + 19T^{2} \) |
| 23 | \( 1 - 2.22T + 23T^{2} \) |
| 29 | \( 1 + 2.40T + 29T^{2} \) |
| 31 | \( 1 - 4.76T + 31T^{2} \) |
| 37 | \( 1 + 9.04T + 37T^{2} \) |
| 41 | \( 1 + 3.19T + 41T^{2} \) |
| 43 | \( 1 - 5.30T + 43T^{2} \) |
| 47 | \( 1 - 5.72T + 47T^{2} \) |
| 53 | \( 1 - 0.714T + 53T^{2} \) |
| 59 | \( 1 + 6.45T + 59T^{2} \) |
| 61 | \( 1 + 13.7T + 61T^{2} \) |
| 67 | \( 1 - 6.34T + 67T^{2} \) |
| 71 | \( 1 - 4.19T + 71T^{2} \) |
| 73 | \( 1 - 7.45T + 73T^{2} \) |
| 79 | \( 1 + 2.56T + 79T^{2} \) |
| 83 | \( 1 - 9.66T + 83T^{2} \) |
| 89 | \( 1 + 14.9T + 89T^{2} \) |
| 97 | \( 1 + 5.33T + 97T^{2} \) |
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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
−7.31021038083685283983249275852, −6.72360252879667553829941782355, −6.23339972222201070911219003941, −5.67235978200643198686870980036, −4.87416793960067251543025638885, −3.64156441898019884340467813304, −2.81108565166411356919312581933, −2.24758000337136009676188793635, −1.25098667036587050015455606038, 0,
1.25098667036587050015455606038, 2.24758000337136009676188793635, 2.81108565166411356919312581933, 3.64156441898019884340467813304, 4.87416793960067251543025638885, 5.67235978200643198686870980036, 6.23339972222201070911219003941, 6.72360252879667553829941782355, 7.31021038083685283983249275852