| L(s) = 1 | + 2-s + (1 − 1.41i)3-s + 4-s − i·5-s + (1 − 1.41i)6-s + 3.41·7-s + 8-s + (−1.00 − 2.82i)9-s − i·10-s + 2.58i·11-s + (1 − 1.41i)12-s + 6.24i·13-s + 3.41·14-s + (−1.41 − i)15-s + 16-s − 2.82i·17-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + (0.577 − 0.816i)3-s + 0.5·4-s − 0.447i·5-s + (0.408 − 0.577i)6-s + 1.29·7-s + 0.353·8-s + (−0.333 − 0.942i)9-s − 0.316i·10-s + 0.779i·11-s + (0.288 − 0.408i)12-s + 1.73i·13-s + 0.912·14-s + (−0.365 − 0.258i)15-s + 0.250·16-s − 0.685i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.662 + 0.749i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.662 + 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.62226 - 1.18198i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.62226 - 1.18198i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1 + 1.41i)T \) |
| 5 | \( 1 + iT \) |
| 19 | \( 1 + (4.24 + i)T \) |
| good | 7 | \( 1 - 3.41T + 7T^{2} \) |
| 11 | \( 1 - 2.58iT - 11T^{2} \) |
| 13 | \( 1 - 6.24iT - 13T^{2} \) |
| 17 | \( 1 + 2.82iT - 17T^{2} \) |
| 23 | \( 1 + 6iT - 23T^{2} \) |
| 29 | \( 1 + 9.07T + 29T^{2} \) |
| 31 | \( 1 + 1.17iT - 31T^{2} \) |
| 37 | \( 1 - 6.24iT - 37T^{2} \) |
| 41 | \( 1 - 3.07T + 41T^{2} \) |
| 43 | \( 1 - 9.41T + 43T^{2} \) |
| 47 | \( 1 - 8.82iT - 47T^{2} \) |
| 53 | \( 1 + 8.82T + 53T^{2} \) |
| 59 | \( 1 + 7.17T + 59T^{2} \) |
| 61 | \( 1 - 12.4T + 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 - 2.82T + 71T^{2} \) |
| 73 | \( 1 + 14.4T + 73T^{2} \) |
| 79 | \( 1 - 9.31iT - 79T^{2} \) |
| 83 | \( 1 + 7.17iT - 83T^{2} \) |
| 89 | \( 1 - 13.4T + 89T^{2} \) |
| 97 | \( 1 + 9.41iT - 97T^{2} \) |
| 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
−11.05108002096905571874987904490, −9.499443160654880028387070724353, −8.769348124728906008607094931375, −7.80202242227694047962297532147, −7.05893811535262297355140259962, −6.13802179362875063779122290392, −4.72734722309184612335936219697, −4.19930328807414538843215952970, −2.39148028283665903808328174644, −1.62043900200395382227800313105,
2.02476914482624975678854849834, 3.29687294806212818937218886176, 4.06691870951529108305198035204, 5.33073275625658685155226248282, 5.82422646652259103790099859648, 7.60048286717238886568321299370, 8.028681042558206351001899027854, 9.028307458576242894665517722794, 10.32683681460412669327763896967, 10.88839689649656586309831289134
