| L(s) = 1 | + 2-s + 4-s − 5-s + 5.20·7-s + 8-s − 10-s − 4.55·11-s + 1.65·13-s + 5.20·14-s + 16-s − 1.65·17-s + 4·19-s − 20-s − 4.55·22-s − 23-s − 4·25-s + 1.65·26-s + 5.20·28-s + 0.348·29-s + 8.55·31-s + 32-s − 1.65·34-s − 5.20·35-s + 5·37-s + 4·38-s − 40-s + 9.85·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.447·5-s + 1.96·7-s + 0.353·8-s − 0.316·10-s − 1.37·11-s + 0.458·13-s + 1.39·14-s + 0.250·16-s − 0.400·17-s + 0.917·19-s − 0.223·20-s − 0.971·22-s − 0.208·23-s − 0.800·25-s + 0.323·26-s + 0.983·28-s + 0.0646·29-s + 1.53·31-s + 0.176·32-s − 0.283·34-s − 0.880·35-s + 0.821·37-s + 0.648·38-s − 0.158·40-s + 1.53·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3726 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3726 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.478109094\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.478109094\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 + T + 5T^{2} \) |
| 7 | \( 1 - 5.20T + 7T^{2} \) |
| 11 | \( 1 + 4.55T + 11T^{2} \) |
| 13 | \( 1 - 1.65T + 13T^{2} \) |
| 17 | \( 1 + 1.65T + 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 29 | \( 1 - 0.348T + 29T^{2} \) |
| 31 | \( 1 - 8.55T + 31T^{2} \) |
| 37 | \( 1 - 5T + 37T^{2} \) |
| 41 | \( 1 - 9.85T + 41T^{2} \) |
| 43 | \( 1 - 7.20T + 43T^{2} \) |
| 47 | \( 1 + 3.20T + 47T^{2} \) |
| 53 | \( 1 - 13.8T + 53T^{2} \) |
| 59 | \( 1 + 9.20T + 59T^{2} \) |
| 61 | \( 1 + 11.4T + 61T^{2} \) |
| 67 | \( 1 - 1.34T + 67T^{2} \) |
| 71 | \( 1 - 5.34T + 71T^{2} \) |
| 73 | \( 1 + 16.6T + 73T^{2} \) |
| 79 | \( 1 + 9.20T + 79T^{2} \) |
| 83 | \( 1 - 0.555T + 83T^{2} \) |
| 89 | \( 1 - 3.65T + 89T^{2} \) |
| 97 | \( 1 + 6.41T + 97T^{2} \) |
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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
−8.153589380932968717865312940572, −7.80252265605497004072633261108, −7.29644559616147346091174113829, −5.96041635020207621777721128837, −5.47390385312198467130568810612, −4.53478326927608197375631632548, −4.28297986797263372290477024974, −2.93987483571308360657662911994, −2.16588030145832798831058248714, −1.03249712198989284812680034998,
1.03249712198989284812680034998, 2.16588030145832798831058248714, 2.93987483571308360657662911994, 4.28297986797263372290477024974, 4.53478326927608197375631632548, 5.47390385312198467130568810612, 5.96041635020207621777721128837, 7.29644559616147346091174113829, 7.80252265605497004072633261108, 8.153589380932968717865312940572
