L(s) = 1 | − 2.89·3-s + 3.16·5-s − 0.0867·7-s + 5.36·9-s + 1.65·11-s − 1.35·13-s − 9.14·15-s + 3.42·17-s + 0.324·19-s + 0.250·21-s − 8.51·23-s + 5.00·25-s − 6.84·27-s − 8.13·29-s − 6.31·31-s − 4.77·33-s − 0.274·35-s + 0.255·37-s + 3.90·39-s + 3.09·41-s − 4.19·43-s + 16.9·45-s − 0.838·47-s − 6.99·49-s − 9.91·51-s + 10.3·53-s + 5.22·55-s + ⋯ |
L(s) = 1 | − 1.66·3-s + 1.41·5-s − 0.0327·7-s + 1.78·9-s + 0.497·11-s − 0.374·13-s − 2.36·15-s + 0.831·17-s + 0.0745·19-s + 0.0547·21-s − 1.77·23-s + 1.00·25-s − 1.31·27-s − 1.51·29-s − 1.13·31-s − 0.830·33-s − 0.0463·35-s + 0.0420·37-s + 0.625·39-s + 0.483·41-s − 0.639·43-s + 2.53·45-s − 0.122·47-s − 0.998·49-s − 1.38·51-s + 1.42·53-s + 0.703·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 503 | \( 1 + T \) |
good | 3 | \( 1 + 2.89T + 3T^{2} \) |
| 5 | \( 1 - 3.16T + 5T^{2} \) |
| 7 | \( 1 + 0.0867T + 7T^{2} \) |
| 11 | \( 1 - 1.65T + 11T^{2} \) |
| 13 | \( 1 + 1.35T + 13T^{2} \) |
| 17 | \( 1 - 3.42T + 17T^{2} \) |
| 19 | \( 1 - 0.324T + 19T^{2} \) |
| 23 | \( 1 + 8.51T + 23T^{2} \) |
| 29 | \( 1 + 8.13T + 29T^{2} \) |
| 31 | \( 1 + 6.31T + 31T^{2} \) |
| 37 | \( 1 - 0.255T + 37T^{2} \) |
| 41 | \( 1 - 3.09T + 41T^{2} \) |
| 43 | \( 1 + 4.19T + 43T^{2} \) |
| 47 | \( 1 + 0.838T + 47T^{2} \) |
| 53 | \( 1 - 10.3T + 53T^{2} \) |
| 59 | \( 1 + 7.20T + 59T^{2} \) |
| 61 | \( 1 + 14.0T + 61T^{2} \) |
| 67 | \( 1 - 3.88T + 67T^{2} \) |
| 71 | \( 1 - 0.709T + 71T^{2} \) |
| 73 | \( 1 + 5.35T + 73T^{2} \) |
| 79 | \( 1 - 6.01T + 79T^{2} \) |
| 83 | \( 1 + 4.45T + 83T^{2} \) |
| 89 | \( 1 + 3.14T + 89T^{2} \) |
| 97 | \( 1 - 12.9T + 97T^{2} \) |
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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
−7.86058675992442422427499071180, −7.12009664810875844957777167327, −6.28192605630801080210083011956, −5.82266605729403717756176265572, −5.42069223194616852534245459065, −4.55084385373825514799223571168, −3.56518518095070166000309151018, −2.07634666591785978332282045381, −1.39652520189974452384154703911, 0,
1.39652520189974452384154703911, 2.07634666591785978332282045381, 3.56518518095070166000309151018, 4.55084385373825514799223571168, 5.42069223194616852534245459065, 5.82266605729403717756176265572, 6.28192605630801080210083011956, 7.12009664810875844957777167327, 7.86058675992442422427499071180