L(s) = 1 | + i·2-s + i·3-s − 4-s + (2 − i)5-s − 6-s − i·8-s − 9-s + (1 + 2i)10-s − 5·11-s − i·12-s + i·13-s + (1 + 2i)15-s + 16-s + 2i·17-s − i·18-s − 7·19-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + 0.577i·3-s − 0.5·4-s + (0.894 − 0.447i)5-s − 0.408·6-s − 0.353i·8-s − 0.333·9-s + (0.316 + 0.632i)10-s − 1.50·11-s − 0.288i·12-s + 0.277i·13-s + (0.258 + 0.516i)15-s + 0.250·16-s + 0.485i·17-s − 0.235i·18-s − 1.60·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6126547694\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6126547694\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 + (-2 + i)T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + 5T + 11T^{2} \) |
| 13 | \( 1 - iT - 13T^{2} \) |
| 17 | \( 1 - 2iT - 17T^{2} \) |
| 19 | \( 1 + 7T + 19T^{2} \) |
| 23 | \( 1 - 3iT - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 6T + 31T^{2} \) |
| 37 | \( 1 - 5iT - 37T^{2} \) |
| 41 | \( 1 + 9T + 41T^{2} \) |
| 43 | \( 1 - 10iT - 43T^{2} \) |
| 47 | \( 1 - 13iT - 47T^{2} \) |
| 53 | \( 1 + iT - 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 6iT - 67T^{2} \) |
| 71 | \( 1 + 2T + 71T^{2} \) |
| 73 | \( 1 - 4iT - 73T^{2} \) |
| 79 | \( 1 - 14T + 79T^{2} \) |
| 83 | \( 1 + 10iT - 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 + 8iT - 97T^{2} \) |
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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
−9.886854455635435559973941916322, −9.105756100425484653667666449819, −8.388886862842258079571312699229, −7.70976505900251366710457066509, −6.48564632037147651937050486620, −5.86435158814388079943570036739, −5.02093583675257517395915830129, −4.42043626204439815208479484311, −3.07332614542698077553300296824, −1.83193334314084235248106097083,
0.21987276389889351163085446506, 2.00821111096943333565913103016, 2.47132488234768424377477649349, 3.59052577848097768589292253766, 5.00908477864696606905056858633, 5.60923192592448998276820419979, 6.61853468352170945919737778273, 7.42151430090843061699936346803, 8.411092767280935435889115081642, 9.027115059790585158195686307713