L(s) = 1 | − 2.23i·5-s − 0.856i·7-s + 3·9-s − 3.31·11-s − 6.26i·13-s − 6.87·17-s − 5.00·25-s + 8.94i·31-s − 1.91·35-s − 8.52·43-s − 6.70i·45-s + 6.26·49-s + 7.41i·55-s + 4·59-s − 2.56i·63-s + ⋯ |
L(s) = 1 | − 0.999i·5-s − 0.323i·7-s + 9-s − 1.00·11-s − 1.73i·13-s − 1.66·17-s − 1.00·25-s + 1.60i·31-s − 0.323·35-s − 1.30·43-s − 0.999i·45-s + 0.895·49-s + 0.999i·55-s + 0.520·59-s − 0.323i·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.956 + 0.292i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.956 + 0.292i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8859473551\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8859473551\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + 2.23iT \) |
| 11 | \( 1 + 3.31T \) |
good | 3 | \( 1 - 3T^{2} \) |
| 7 | \( 1 + 0.856iT - 7T^{2} \) |
| 13 | \( 1 + 6.26iT - 13T^{2} \) |
| 17 | \( 1 + 6.87T + 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 8.94iT - 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + 8.52T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 14.8iT - 71T^{2} \) |
| 73 | \( 1 + 0.261T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 12.3T + 83T^{2} \) |
| 89 | \( 1 + 13.2T + 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−8.802670605893892452985457418198, −8.213925067480358185159449662025, −7.46549115510078380681632237868, −6.62774249918550482989887327133, −5.44372559207407110978999666040, −4.90172501668978587787527546934, −4.05444045937019650046913760055, −2.87471986164027152238171147870, −1.59319850298194040195884928345, −0.31398047688048874949246405150,
1.93537516597041779797596231943, 2.54979581727447294303469569823, 3.96145804511360078467476499246, 4.52673879180040770617173347153, 5.73806322219897455995689336941, 6.76735889138918967583139273317, 6.99956739210711592074431018271, 8.013161605605391982102002684666, 8.952559202431732976899229237467, 9.730026163707198139688429368555