L(s) = 1 | + (0.382 − 0.662i)2-s + (0.5 + 0.866i)3-s + (0.207 + 0.358i)4-s + (−0.923 + 1.60i)5-s + 0.765·6-s + 1.08·8-s + (−0.499 + 0.866i)9-s + (0.707 + 1.22i)10-s + (−0.382 − 0.662i)11-s + (−0.207 + 0.358i)12-s + 13-s − 1.84·15-s + (0.207 − 0.358i)16-s + (0.382 + 0.662i)18-s − 0.765·20-s + ⋯ |
L(s) = 1 | + (0.382 − 0.662i)2-s + (0.5 + 0.866i)3-s + (0.207 + 0.358i)4-s + (−0.923 + 1.60i)5-s + 0.765·6-s + 1.08·8-s + (−0.499 + 0.866i)9-s + (0.707 + 1.22i)10-s + (−0.382 − 0.662i)11-s + (−0.207 + 0.358i)12-s + 13-s − 1.84·15-s + (0.207 − 0.358i)16-s + (0.382 + 0.662i)18-s − 0.765·20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0725 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0725 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.558594963\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.558594963\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.5 - 0.866i)T \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 2 | \( 1 + (-0.382 + 0.662i)T + (-0.5 - 0.866i)T^{2} \) |
| 5 | \( 1 + (0.923 - 1.60i)T + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (0.382 + 0.662i)T + (-0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 + 1.84T + T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + (-0.923 + 1.60i)T + (-0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (-0.382 - 0.662i)T + (-0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 - 1.84T + T^{2} \) |
| 73 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 - 0.765T + T^{2} \) |
| 89 | \( 1 + (0.382 - 0.662i)T + (-0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.985076498480109555091633703864, −8.577497780809319458182342626991, −8.158112192549490326114139553098, −7.30865768036459779459515924652, −6.56321056836476419475595980921, −5.37113145426023027804229118592, −4.13360814773022066647775415400, −3.58717011386118334780769622848, −3.08452602363628122711101403439, −2.23034998956382205137052800915,
1.01463037107821888795359710354, 1.91933131029070183671142782695, 3.54053410250692369725038128313, 4.46443692228973926134222234103, 5.20988683490269800790658835907, 6.04734493830427450710061232547, 6.93771223130469718979244866509, 7.70227339035689055620140766771, 8.207434843465743563140701672044, 8.892114472252092552993540934419