L(s) = 1 | − 2.19i·2-s − i·3-s − 2.83·4-s + i·5-s − 2.19·6-s + (2.19 + 1.47i)7-s + 1.83i·8-s − 9-s + 2.19·10-s + (−1.23 + 3.07i)11-s + 2.83i·12-s − 6.52·13-s + (3.23 − 4.83i)14-s + 15-s − 1.63·16-s − 3.82·17-s + ⋯ |
L(s) = 1 | − 1.55i·2-s − 0.577i·3-s − 1.41·4-s + 0.447i·5-s − 0.897·6-s + (0.830 + 0.556i)7-s + 0.647i·8-s − 0.333·9-s + 0.695·10-s + (−0.372 + 0.927i)11-s + 0.817i·12-s − 1.80·13-s + (0.864 − 1.29i)14-s + 0.258·15-s − 0.409·16-s − 0.927·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.825 - 0.563i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.825 - 0.563i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3845906191\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3845906191\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + iT \) |
| 5 | \( 1 - iT \) |
| 7 | \( 1 + (-2.19 - 1.47i)T \) |
| 11 | \( 1 + (1.23 - 3.07i)T \) |
good | 2 | \( 1 + 2.19iT - 2T^{2} \) |
| 13 | \( 1 + 6.52T + 13T^{2} \) |
| 17 | \( 1 + 3.82T + 17T^{2} \) |
| 19 | \( 1 + 2.33T + 19T^{2} \) |
| 23 | \( 1 - 3.83T + 23T^{2} \) |
| 29 | \( 1 - 4.90iT - 29T^{2} \) |
| 31 | \( 1 - 7.93iT - 31T^{2} \) |
| 37 | \( 1 + 7.93T + 37T^{2} \) |
| 41 | \( 1 + 10.1T + 41T^{2} \) |
| 43 | \( 1 + 6.37iT - 43T^{2} \) |
| 47 | \( 1 + 2.06iT - 47T^{2} \) |
| 53 | \( 1 - 10.0T + 53T^{2} \) |
| 59 | \( 1 + 3.76iT - 59T^{2} \) |
| 61 | \( 1 + 9.22T + 61T^{2} \) |
| 67 | \( 1 + 2.13T + 67T^{2} \) |
| 71 | \( 1 - 8.40T + 71T^{2} \) |
| 73 | \( 1 + 8.34T + 73T^{2} \) |
| 79 | \( 1 + 13.1iT - 79T^{2} \) |
| 83 | \( 1 - 15.3T + 83T^{2} \) |
| 89 | \( 1 - 0.705iT - 89T^{2} \) |
| 97 | \( 1 + 10.0iT - 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.28949682995894465566752433091, −9.143551881692519017200596371226, −8.573050453430060697754332388966, −7.26689329095655590063115327784, −6.86193704873139729233109175174, −5.10867863633406971849486272598, −4.72532142286426885183731439962, −3.24162384831943992904045908531, −2.27626766500729590777957418121, −1.80131268849815021044891662467,
0.15331985301659259996622565080, 2.39923891625424883576567120193, 4.09714359845064746818476562851, 4.86080699765726217385739315541, 5.33203270579584884886192483676, 6.40702376949360569948318692937, 7.31321578558022295019934097388, 7.998706089100709178945795025553, 8.649602763612048252953061508234, 9.407828941515321332685326117083