L(s) = 1 | + (0.951 − 0.690i)2-s + (0.309 + 0.951i)3-s + (0.118 − 0.363i)4-s + (0.951 + 0.690i)6-s + (0.224 + 0.690i)8-s + (−0.809 + 0.587i)9-s + (−0.951 − 0.309i)11-s + 0.381·12-s + (−1.30 + 0.951i)13-s + (0.999 + 0.726i)16-s + (1.53 + 1.11i)17-s + (−0.363 + 1.11i)18-s + (−0.190 − 0.587i)19-s + (−1.11 + 0.363i)22-s + 1.17·23-s + (−0.587 + 0.427i)24-s + ⋯ |
L(s) = 1 | + (0.951 − 0.690i)2-s + (0.309 + 0.951i)3-s + (0.118 − 0.363i)4-s + (0.951 + 0.690i)6-s + (0.224 + 0.690i)8-s + (−0.809 + 0.587i)9-s + (−0.951 − 0.309i)11-s + 0.381·12-s + (−1.30 + 0.951i)13-s + (0.999 + 0.726i)16-s + (1.53 + 1.11i)17-s + (−0.363 + 1.11i)18-s + (−0.190 − 0.587i)19-s + (−1.11 + 0.363i)22-s + 1.17·23-s + (−0.587 + 0.427i)24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.606 - 0.794i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.606 - 0.794i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.880151548\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.880151548\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.309 - 0.951i)T \) |
| 11 | \( 1 + (0.951 + 0.309i)T \) |
| 61 | \( 1 + (0.809 + 0.587i)T \) |
good | 2 | \( 1 + (-0.951 + 0.690i)T + (0.309 - 0.951i)T^{2} \) |
| 5 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 7 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 13 | \( 1 + (1.30 - 0.951i)T + (0.309 - 0.951i)T^{2} \) |
| 17 | \( 1 + (-1.53 - 1.11i)T + (0.309 + 0.951i)T^{2} \) |
| 19 | \( 1 + (0.190 + 0.587i)T + (-0.809 + 0.587i)T^{2} \) |
| 23 | \( 1 - 1.17T + T^{2} \) |
| 29 | \( 1 + (-0.363 + 1.11i)T + (-0.809 - 0.587i)T^{2} \) |
| 31 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 37 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 41 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 53 | \( 1 + (-0.951 + 0.690i)T + (0.309 - 0.951i)T^{2} \) |
| 59 | \( 1 + (-0.587 + 1.80i)T + (-0.809 - 0.587i)T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 73 | \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \) |
| 79 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 83 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 89 | \( 1 + 1.90T + T^{2} \) |
| 97 | \( 1 + (-1.61 + 1.17i)T + (0.309 - 0.951i)T^{2} \) |
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
−9.778482119525823529641886973746, −8.686741184184253810932709864641, −8.061986046054118776069476921704, −7.18591952273659608397545000045, −5.79349997342300885457253827308, −5.09445211914810150956158981878, −4.57863062740336700819655555089, −3.58129980273486191185800113112, −2.93759876269339867738022719665, −2.04808198110323788135765302906,
0.996581477392873738084179889530, 2.69766916878442046934255935326, 3.19520959537857469878180346270, 4.70266578900420771748680034728, 5.30608853405786440273910954901, 5.92182996429275914723728966238, 7.03635747361293611049415294272, 7.42892517654464261176657617476, 7.989464252516109076573823353005, 9.104255890087660990882493958621