| L(s) = 1 | + (2.10 − 0.754i)5-s − 1.11i·7-s − 0.423i·11-s + 3.69i·13-s + 6.28·17-s − 3.85i·19-s + 3.79i·23-s + (3.86 − 3.17i)25-s + (4.89 + 2.25i)29-s + 7.17i·31-s + (−0.838 − 2.33i)35-s − 5.33·37-s − 6.08i·41-s + 7.16·43-s − 8.32·47-s + ⋯ |
| L(s) = 1 | + (0.941 − 0.337i)5-s − 0.419i·7-s − 0.127i·11-s + 1.02i·13-s + 1.52·17-s − 0.884i·19-s + 0.790i·23-s + (0.772 − 0.635i)25-s + (0.908 + 0.418i)29-s + 1.28i·31-s + (−0.141 − 0.395i)35-s − 0.877·37-s − 0.950i·41-s + 1.09·43-s − 1.21·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0876i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5220 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 + 0.0876i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.609349557\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.609349557\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-2.10 + 0.754i)T \) |
| 29 | \( 1 + (-4.89 - 2.25i)T \) |
| good | 7 | \( 1 + 1.11iT - 7T^{2} \) |
| 11 | \( 1 + 0.423iT - 11T^{2} \) |
| 13 | \( 1 - 3.69iT - 13T^{2} \) |
| 17 | \( 1 - 6.28T + 17T^{2} \) |
| 19 | \( 1 + 3.85iT - 19T^{2} \) |
| 23 | \( 1 - 3.79iT - 23T^{2} \) |
| 31 | \( 1 - 7.17iT - 31T^{2} \) |
| 37 | \( 1 + 5.33T + 37T^{2} \) |
| 41 | \( 1 + 6.08iT - 41T^{2} \) |
| 43 | \( 1 - 7.16T + 43T^{2} \) |
| 47 | \( 1 + 8.32T + 47T^{2} \) |
| 53 | \( 1 - 7.82iT - 53T^{2} \) |
| 59 | \( 1 + 1.94T + 59T^{2} \) |
| 61 | \( 1 + 12.8iT - 61T^{2} \) |
| 67 | \( 1 - 14.8iT - 67T^{2} \) |
| 71 | \( 1 - 3.62T + 71T^{2} \) |
| 73 | \( 1 - 1.15T + 73T^{2} \) |
| 79 | \( 1 + 3.85iT - 79T^{2} \) |
| 83 | \( 1 - 6.56iT - 83T^{2} \) |
| 89 | \( 1 - 11.4iT - 89T^{2} \) |
| 97 | \( 1 - 0.415T + 97T^{2} \) |
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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
−8.332133466067022974665199878201, −7.28520959245692278422188273576, −6.85800603167105139643281447240, −5.97990148115754995514273495260, −5.27530056639087176180903068799, −4.66976671875692359153524942293, −3.65963370365861631914379523980, −2.82065675783613976838881816466, −1.73134738001626759340979318171, −0.964195743213655488433613311224,
0.878186869662910901624670873827, 1.98277878596190805451522225653, 2.83375462725186491103520622587, 3.51608409864221864872703950309, 4.68138497941698840754776358078, 5.53021612858847755121669015934, 5.93690238404702933170577495583, 6.62091274991513879728919819642, 7.66886740471743536678728398708, 8.083288129618822472746485956517
