| L(s) = 1 | − 1.16·2-s − 2.59·3-s − 0.648·4-s − 0.596·5-s + 3.01·6-s − 2.62·7-s + 3.07·8-s + 3.71·9-s + 0.693·10-s − 11-s + 1.67·12-s − 0.231·13-s + 3.05·14-s + 1.54·15-s − 2.28·16-s + 17-s − 4.31·18-s + 5.45·19-s + 0.386·20-s + 6.79·21-s + 1.16·22-s − 0.801·23-s − 7.97·24-s − 4.64·25-s + 0.269·26-s − 1.84·27-s + 1.70·28-s + ⋯ |
| L(s) = 1 | − 0.822·2-s − 1.49·3-s − 0.324·4-s − 0.266·5-s + 1.22·6-s − 0.991·7-s + 1.08·8-s + 1.23·9-s + 0.219·10-s − 0.301·11-s + 0.484·12-s − 0.0641·13-s + 0.815·14-s + 0.398·15-s − 0.570·16-s + 0.242·17-s − 1.01·18-s + 1.25·19-s + 0.0863·20-s + 1.48·21-s + 0.247·22-s − 0.167·23-s − 1.62·24-s − 0.928·25-s + 0.0527·26-s − 0.355·27-s + 0.321·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4748856318\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4748856318\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 43 | \( 1 + T \) |
| good | 2 | \( 1 + 1.16T + 2T^{2} \) |
| 3 | \( 1 + 2.59T + 3T^{2} \) |
| 5 | \( 1 + 0.596T + 5T^{2} \) |
| 7 | \( 1 + 2.62T + 7T^{2} \) |
| 13 | \( 1 + 0.231T + 13T^{2} \) |
| 19 | \( 1 - 5.45T + 19T^{2} \) |
| 23 | \( 1 + 0.801T + 23T^{2} \) |
| 29 | \( 1 - 7.41T + 29T^{2} \) |
| 31 | \( 1 - 7.39T + 31T^{2} \) |
| 37 | \( 1 - 7.80T + 37T^{2} \) |
| 41 | \( 1 - 8.33T + 41T^{2} \) |
| 47 | \( 1 - 6.73T + 47T^{2} \) |
| 53 | \( 1 + 5.75T + 53T^{2} \) |
| 59 | \( 1 - 2.56T + 59T^{2} \) |
| 61 | \( 1 - 1.46T + 61T^{2} \) |
| 67 | \( 1 - 6.54T + 67T^{2} \) |
| 71 | \( 1 + 8.44T + 71T^{2} \) |
| 73 | \( 1 + 5.18T + 73T^{2} \) |
| 79 | \( 1 - 10.3T + 79T^{2} \) |
| 83 | \( 1 - 9.24T + 83T^{2} \) |
| 89 | \( 1 - 14.9T + 89T^{2} \) |
| 97 | \( 1 + 1.30T + 97T^{2} \) |
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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
−7.74596022714389063949827336502, −7.24810441106398076608755879194, −6.30468752747292850182102742219, −5.96161511722091678001427786581, −5.02025579502674453658647695684, −4.53143168218017527152415045074, −3.62159569476314265742051544262, −2.58345597208477828408332136136, −1.06555097602854219092499140143, −0.54148382608860032405032352292,
0.54148382608860032405032352292, 1.06555097602854219092499140143, 2.58345597208477828408332136136, 3.62159569476314265742051544262, 4.53143168218017527152415045074, 5.02025579502674453658647695684, 5.96161511722091678001427786581, 6.30468752747292850182102742219, 7.24810441106398076608755879194, 7.74596022714389063949827336502
