| L(s) = 1 | − 2.29·3-s − 3.92·5-s + 2.25·9-s + 3.51·11-s − 3.61·13-s + 8.99·15-s − 6.69·17-s + 0.207·19-s − 2.79·23-s + 10.3·25-s + 1.71·27-s − 2.98·29-s + 4.29·31-s − 8.05·33-s − 10.2·37-s + 8.28·39-s − 41-s + 9.05·43-s − 8.83·45-s − 10.5·47-s + 15.3·51-s − 12.7·53-s − 13.7·55-s − 0.475·57-s + 8.12·59-s + 3.81·61-s + 14.1·65-s + ⋯ |
| L(s) = 1 | − 1.32·3-s − 1.75·5-s + 0.750·9-s + 1.05·11-s − 1.00·13-s + 2.32·15-s − 1.62·17-s + 0.0475·19-s − 0.581·23-s + 2.07·25-s + 0.329·27-s − 0.554·29-s + 0.771·31-s − 1.40·33-s − 1.69·37-s + 1.32·39-s − 0.156·41-s + 1.38·43-s − 1.31·45-s − 1.53·47-s + 2.14·51-s − 1.74·53-s − 1.85·55-s − 0.0629·57-s + 1.05·59-s + 0.488·61-s + 1.75·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.08411811999\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.08411811999\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 3 | \( 1 + 2.29T + 3T^{2} \) |
| 5 | \( 1 + 3.92T + 5T^{2} \) |
| 11 | \( 1 - 3.51T + 11T^{2} \) |
| 13 | \( 1 + 3.61T + 13T^{2} \) |
| 17 | \( 1 + 6.69T + 17T^{2} \) |
| 19 | \( 1 - 0.207T + 19T^{2} \) |
| 23 | \( 1 + 2.79T + 23T^{2} \) |
| 29 | \( 1 + 2.98T + 29T^{2} \) |
| 31 | \( 1 - 4.29T + 31T^{2} \) |
| 37 | \( 1 + 10.2T + 37T^{2} \) |
| 43 | \( 1 - 9.05T + 43T^{2} \) |
| 47 | \( 1 + 10.5T + 47T^{2} \) |
| 53 | \( 1 + 12.7T + 53T^{2} \) |
| 59 | \( 1 - 8.12T + 59T^{2} \) |
| 61 | \( 1 - 3.81T + 61T^{2} \) |
| 67 | \( 1 + 4.59T + 67T^{2} \) |
| 71 | \( 1 + 5.96T + 71T^{2} \) |
| 73 | \( 1 + 14.0T + 73T^{2} \) |
| 79 | \( 1 + 12.3T + 79T^{2} \) |
| 83 | \( 1 + 5.55T + 83T^{2} \) |
| 89 | \( 1 - 2.33T + 89T^{2} \) |
| 97 | \( 1 + 8.84T + 97T^{2} \) |
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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
−7.67978374203998690546158073338, −6.90896273584871254780586049576, −6.69346421437371367298476916581, −5.76903836554721871669780612543, −4.82179713806364894576632833431, −4.42893010897636215933738280777, −3.80540649011417440116491408649, −2.79244400503393150873403547793, −1.46455624237682239063215166917, −0.15840140354290704407227158277,
0.15840140354290704407227158277, 1.46455624237682239063215166917, 2.79244400503393150873403547793, 3.80540649011417440116491408649, 4.42893010897636215933738280777, 4.82179713806364894576632833431, 5.76903836554721871669780612543, 6.69346421437371367298476916581, 6.90896273584871254780586049576, 7.67978374203998690546158073338
