| L(s) = 1 | + 0.981·2-s + 1.43·3-s − 1.03·4-s − 0.0504·5-s + 1.41·6-s − 1.94·7-s − 2.98·8-s − 0.932·9-s − 0.0494·10-s + 0.678·11-s − 1.48·12-s − 1.90·14-s − 0.0724·15-s − 0.854·16-s + 5.99·17-s − 0.915·18-s + 6.66·19-s + 0.0522·20-s − 2.79·21-s + 0.665·22-s + 4.71·23-s − 4.28·24-s − 4.99·25-s − 5.65·27-s + 2.01·28-s − 6.85·29-s − 0.0711·30-s + ⋯ |
| L(s) = 1 | + 0.694·2-s + 0.830·3-s − 0.518·4-s − 0.0225·5-s + 0.576·6-s − 0.735·7-s − 1.05·8-s − 0.310·9-s − 0.0156·10-s + 0.204·11-s − 0.430·12-s − 0.510·14-s − 0.0187·15-s − 0.213·16-s + 1.45·17-s − 0.215·18-s + 1.52·19-s + 0.0116·20-s − 0.610·21-s + 0.141·22-s + 0.983·23-s − 0.874·24-s − 0.999·25-s − 1.08·27-s + 0.380·28-s − 1.27·29-s − 0.0129·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5239 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5239 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 31 | \( 1 + T \) |
| good | 2 | \( 1 - 0.981T + 2T^{2} \) |
| 3 | \( 1 - 1.43T + 3T^{2} \) |
| 5 | \( 1 + 0.0504T + 5T^{2} \) |
| 7 | \( 1 + 1.94T + 7T^{2} \) |
| 11 | \( 1 - 0.678T + 11T^{2} \) |
| 17 | \( 1 - 5.99T + 17T^{2} \) |
| 19 | \( 1 - 6.66T + 19T^{2} \) |
| 23 | \( 1 - 4.71T + 23T^{2} \) |
| 29 | \( 1 + 6.85T + 29T^{2} \) |
| 37 | \( 1 + 6.27T + 37T^{2} \) |
| 41 | \( 1 + 2.38T + 41T^{2} \) |
| 43 | \( 1 + 8.25T + 43T^{2} \) |
| 47 | \( 1 - 10.6T + 47T^{2} \) |
| 53 | \( 1 + 0.411T + 53T^{2} \) |
| 59 | \( 1 - 0.0347T + 59T^{2} \) |
| 61 | \( 1 + 13.4T + 61T^{2} \) |
| 67 | \( 1 + 14.9T + 67T^{2} \) |
| 71 | \( 1 + 3.70T + 71T^{2} \) |
| 73 | \( 1 + 10.9T + 73T^{2} \) |
| 79 | \( 1 + 9.22T + 79T^{2} \) |
| 83 | \( 1 + 4.11T + 83T^{2} \) |
| 89 | \( 1 + 0.657T + 89T^{2} \) |
| 97 | \( 1 - 17.9T + 97T^{2} \) |
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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
−7.67009110114598691184179403948, −7.38398622742393685399300629558, −6.11175661693058789379616854836, −5.62041093306845636367605125438, −4.96506886622945086054265250990, −3.81642045566353553345740776606, −3.30828262025084071605975301486, −2.92351018885498768776639148528, −1.47715421971695612483914183472, 0,
1.47715421971695612483914183472, 2.92351018885498768776639148528, 3.30828262025084071605975301486, 3.81642045566353553345740776606, 4.96506886622945086054265250990, 5.62041093306845636367605125438, 6.11175661693058789379616854836, 7.38398622742393685399300629558, 7.67009110114598691184179403948
