| L(s) = 1 | + 2-s − 2.38·3-s + 4-s + 1.29·5-s − 2.38·6-s − 1.08·7-s + 8-s + 2.68·9-s + 1.29·10-s − 4.97·11-s − 2.38·12-s − 5.08·13-s − 1.08·14-s − 3.07·15-s + 16-s − 7.42·17-s + 2.68·18-s − 1.85·19-s + 1.29·20-s + 2.58·21-s − 4.97·22-s − 0.0220·23-s − 2.38·24-s − 3.33·25-s − 5.08·26-s + 0.745·27-s − 1.08·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.37·3-s + 0.5·4-s + 0.577·5-s − 0.973·6-s − 0.408·7-s + 0.353·8-s + 0.895·9-s + 0.408·10-s − 1.50·11-s − 0.688·12-s − 1.41·13-s − 0.289·14-s − 0.794·15-s + 0.250·16-s − 1.79·17-s + 0.633·18-s − 0.426·19-s + 0.288·20-s + 0.563·21-s − 1.06·22-s − 0.00459·23-s − 0.486·24-s − 0.666·25-s − 0.998·26-s + 0.143·27-s − 0.204·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8367349745\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8367349745\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 31 | \( 1 - T \) |
| 97 | \( 1 - T \) |
| good | 3 | \( 1 + 2.38T + 3T^{2} \) |
| 5 | \( 1 - 1.29T + 5T^{2} \) |
| 7 | \( 1 + 1.08T + 7T^{2} \) |
| 11 | \( 1 + 4.97T + 11T^{2} \) |
| 13 | \( 1 + 5.08T + 13T^{2} \) |
| 17 | \( 1 + 7.42T + 17T^{2} \) |
| 19 | \( 1 + 1.85T + 19T^{2} \) |
| 23 | \( 1 + 0.0220T + 23T^{2} \) |
| 29 | \( 1 - 2.19T + 29T^{2} \) |
| 37 | \( 1 + 8.42T + 37T^{2} \) |
| 41 | \( 1 - 5.23T + 41T^{2} \) |
| 43 | \( 1 + 3.86T + 43T^{2} \) |
| 47 | \( 1 - 9.43T + 47T^{2} \) |
| 53 | \( 1 - 8.16T + 53T^{2} \) |
| 59 | \( 1 - 5.60T + 59T^{2} \) |
| 61 | \( 1 - 9.36T + 61T^{2} \) |
| 67 | \( 1 - 13.4T + 67T^{2} \) |
| 71 | \( 1 - 11.7T + 71T^{2} \) |
| 73 | \( 1 - 12.9T + 73T^{2} \) |
| 79 | \( 1 + 0.848T + 79T^{2} \) |
| 83 | \( 1 + 6.31T + 83T^{2} \) |
| 89 | \( 1 + 4.25T + 89T^{2} \) |
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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
−7.85823861775598534220244039267, −6.87463762919807804592640008168, −6.67190161039452943853031274637, −5.74808382072307286756306770690, −5.21217510061120373196642757067, −4.80986877725242274234883803178, −3.89569354677229726512856340976, −2.49614040096164382517384358713, −2.22664898767938041959879706431, −0.42603300569646594137775609722,
0.42603300569646594137775609722, 2.22664898767938041959879706431, 2.49614040096164382517384358713, 3.89569354677229726512856340976, 4.80986877725242274234883803178, 5.21217510061120373196642757067, 5.74808382072307286756306770690, 6.67190161039452943853031274637, 6.87463762919807804592640008168, 7.85823861775598534220244039267
