| L(s) = 1 | + 2.53·3-s − 3.75·5-s + 3.41·9-s − 6.45·11-s + 3.57·13-s − 9.51·15-s − 1.34·17-s + 5.71·19-s + 1.83·23-s + 9.12·25-s + 1.04·27-s − 7.06·29-s − 3.51·31-s − 16.3·33-s − 2.05·37-s + 9.04·39-s + 41-s − 7.47·43-s − 12.8·45-s + 8.31·47-s − 3.41·51-s + 3.63·53-s + 24.2·55-s + 14.4·57-s − 5.51·59-s + 13.0·61-s − 13.4·65-s + ⋯ |
| L(s) = 1 | + 1.46·3-s − 1.68·5-s + 1.13·9-s − 1.94·11-s + 0.991·13-s − 2.45·15-s − 0.326·17-s + 1.31·19-s + 0.383·23-s + 1.82·25-s + 0.200·27-s − 1.31·29-s − 0.631·31-s − 2.84·33-s − 0.338·37-s + 1.44·39-s + 0.156·41-s − 1.14·43-s − 1.91·45-s + 1.21·47-s − 0.477·51-s + 0.498·53-s + 3.27·55-s + 1.91·57-s − 0.718·59-s + 1.67·61-s − 1.66·65-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
| \(L(1)\) |
\(\approx\) |
\(1.934179978\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.934179978\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.53T + 3T^{2} \) |
| 5 | \( 1 + 3.75T + 5T^{2} \) |
| 11 | \( 1 + 6.45T + 11T^{2} \) |
| 13 | \( 1 - 3.57T + 13T^{2} \) |
| 17 | \( 1 + 1.34T + 17T^{2} \) |
| 19 | \( 1 - 5.71T + 19T^{2} \) |
| 23 | \( 1 - 1.83T + 23T^{2} \) |
| 29 | \( 1 + 7.06T + 29T^{2} \) |
| 31 | \( 1 + 3.51T + 31T^{2} \) |
| 37 | \( 1 + 2.05T + 37T^{2} \) |
| 43 | \( 1 + 7.47T + 43T^{2} \) |
| 47 | \( 1 - 8.31T + 47T^{2} \) |
| 53 | \( 1 - 3.63T + 53T^{2} \) |
| 59 | \( 1 + 5.51T + 59T^{2} \) |
| 61 | \( 1 - 13.0T + 61T^{2} \) |
| 67 | \( 1 - 1.51T + 67T^{2} \) |
| 71 | \( 1 + 1.14T + 71T^{2} \) |
| 73 | \( 1 - 11.5T + 73T^{2} \) |
| 79 | \( 1 - 15.2T + 79T^{2} \) |
| 83 | \( 1 - 12.1T + 83T^{2} \) |
| 89 | \( 1 - 11.0T + 89T^{2} \) |
| 97 | \( 1 - 8.04T + 97T^{2} \) |
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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
−7.86060661018158290383489986714, −7.52674575625086445526318104450, −6.86452946889045734278285942163, −5.52586420916382386165309134503, −4.95425904708239323460192277839, −3.86463085249833848293347314339, −3.53322516311942860060558968961, −2.89032812791184590673560434094, −2.03223794366855792545218592851, −0.61638799082704095759805039630,
0.61638799082704095759805039630, 2.03223794366855792545218592851, 2.89032812791184590673560434094, 3.53322516311942860060558968961, 3.86463085249833848293347314339, 4.95425904708239323460192277839, 5.52586420916382386165309134503, 6.86452946889045734278285942163, 7.52674575625086445526318104450, 7.86060661018158290383489986714
