| L(s) = 1 | + (1.41 − 1.73i)5-s + (2.44 − i)7-s − 3·9-s − 3.46·11-s + 3.46i·13-s − 4·17-s + 2.82i·19-s − 4.89·23-s + (−0.999 − 4.89i)25-s − 6.92i·29-s − 9.79·31-s + (1.73 − 5.65i)35-s − 5.65·37-s − 9.79i·41-s − 2.82i·43-s + ⋯ |
| L(s) = 1 | + (0.632 − 0.774i)5-s + (0.925 − 0.377i)7-s − 9-s − 1.04·11-s + 0.960i·13-s − 0.970·17-s + 0.648i·19-s − 1.02·23-s + (−0.199 − 0.979i)25-s − 1.28i·29-s − 1.75·31-s + (0.292 − 0.956i)35-s − 0.929·37-s − 1.53i·41-s − 0.431i·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0353i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0353i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3458687555\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3458687555\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-1.41 + 1.73i)T \) |
| 7 | \( 1 + (-2.44 + i)T \) |
| good | 3 | \( 1 + 3T^{2} \) |
| 11 | \( 1 + 3.46T + 11T^{2} \) |
| 13 | \( 1 - 3.46iT - 13T^{2} \) |
| 17 | \( 1 + 4T + 17T^{2} \) |
| 19 | \( 1 - 2.82iT - 19T^{2} \) |
| 23 | \( 1 + 4.89T + 23T^{2} \) |
| 29 | \( 1 + 6.92iT - 29T^{2} \) |
| 31 | \( 1 + 9.79T + 31T^{2} \) |
| 37 | \( 1 + 5.65T + 37T^{2} \) |
| 41 | \( 1 + 9.79iT - 41T^{2} \) |
| 43 | \( 1 + 2.82iT - 43T^{2} \) |
| 47 | \( 1 - 10iT - 47T^{2} \) |
| 53 | \( 1 - 11.3T + 53T^{2} \) |
| 59 | \( 1 - 8.48iT - 59T^{2} \) |
| 61 | \( 1 + 14.1T + 61T^{2} \) |
| 67 | \( 1 - 14.1iT - 67T^{2} \) |
| 71 | \( 1 + 4iT - 71T^{2} \) |
| 73 | \( 1 + 12T + 73T^{2} \) |
| 79 | \( 1 + 12iT - 79T^{2} \) |
| 83 | \( 1 - 13.8T + 83T^{2} \) |
| 89 | \( 1 + 9.79iT - 89T^{2} \) |
| 97 | \( 1 + 4T + 97T^{2} \) |
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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
−8.794889613624229652783425104337, −7.944317051104182891099107956956, −7.26819218517255156485326372238, −5.98173051937992779270794106650, −5.57257343482932371503553052845, −4.64397414718016179760564469170, −3.93454813984973977505399314135, −2.37079370679177016053650770770, −1.77609028711002670821622013689, −0.10391773069478189563154307978,
1.89610849093082700472594525252, 2.62898746711439258320554918303, 3.46478396092198670178042549865, 4.99319090811635573478522527896, 5.40254614601346541952025169551, 6.18635062225754438826724190187, 7.16187646052295505589553876685, 7.937104770834581927163484844378, 8.617722812701910846505185355216, 9.317716764803535026419613365122
