L(s) = 1 | − 2-s − 3-s + 4-s − 1.14·5-s + 6-s − 8-s + 9-s + 1.14·10-s − 2.16·11-s − 12-s + 2.29·13-s + 1.14·15-s + 16-s − 4.99·17-s − 18-s + 19-s − 1.14·20-s + 2.16·22-s − 6.91·23-s + 24-s − 3.67·25-s − 2.29·26-s − 27-s + 3.76·29-s − 1.14·30-s + 4.38·31-s − 32-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.513·5-s + 0.408·6-s − 0.353·8-s + 0.333·9-s + 0.363·10-s − 0.653·11-s − 0.288·12-s + 0.637·13-s + 0.296·15-s + 0.250·16-s − 1.21·17-s − 0.235·18-s + 0.229·19-s − 0.256·20-s + 0.462·22-s − 1.44·23-s + 0.204·24-s − 0.735·25-s − 0.450·26-s − 0.192·27-s + 0.699·29-s − 0.209·30-s + 0.787·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5586 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5586 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5709119188\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5709119188\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 5 | \( 1 + 1.14T + 5T^{2} \) |
| 11 | \( 1 + 2.16T + 11T^{2} \) |
| 13 | \( 1 - 2.29T + 13T^{2} \) |
| 17 | \( 1 + 4.99T + 17T^{2} \) |
| 23 | \( 1 + 6.91T + 23T^{2} \) |
| 29 | \( 1 - 3.76T + 29T^{2} \) |
| 31 | \( 1 - 4.38T + 31T^{2} \) |
| 37 | \( 1 - 0.660T + 37T^{2} \) |
| 41 | \( 1 - 0.806T + 41T^{2} \) |
| 43 | \( 1 + 2.08T + 43T^{2} \) |
| 47 | \( 1 + 3.72T + 47T^{2} \) |
| 53 | \( 1 + 11.2T + 53T^{2} \) |
| 59 | \( 1 - 10.4T + 59T^{2} \) |
| 61 | \( 1 - 2.57T + 61T^{2} \) |
| 67 | \( 1 - 8.88T + 67T^{2} \) |
| 71 | \( 1 + 0.323T + 71T^{2} \) |
| 73 | \( 1 + 4.72T + 73T^{2} \) |
| 79 | \( 1 + 7.59T + 79T^{2} \) |
| 83 | \( 1 - 3.17T + 83T^{2} \) |
| 89 | \( 1 - 12.3T + 89T^{2} \) |
| 97 | \( 1 + 5.91T + 97T^{2} \) |
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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
−8.143705725583116978406217381642, −7.60283553221524167214176877956, −6.64722666643762051813120747443, −6.23583255586284986212626408490, −5.35355714971362669607760477029, −4.46287441055931450496799075880, −3.73914344264506685314323467025, −2.63722560392094282666601242514, −1.70703626830159924131556051716, −0.45559284860522707379766075676,
0.45559284860522707379766075676, 1.70703626830159924131556051716, 2.63722560392094282666601242514, 3.73914344264506685314323467025, 4.46287441055931450496799075880, 5.35355714971362669607760477029, 6.23583255586284986212626408490, 6.64722666643762051813120747443, 7.60283553221524167214176877956, 8.143705725583116978406217381642