| L(s) = 1 | − 2·2-s + 3·3-s + 4·4-s + 21.3·5-s − 6·6-s − 22.7·7-s − 8·8-s + 9·9-s − 42.7·10-s + 54.7·11-s + 12·12-s + 8.10·13-s + 45.4·14-s + 64.1·15-s + 16·16-s + 72.7·17-s − 18·18-s + 85.4·20-s − 68.2·21-s − 109.·22-s − 27.3·23-s − 24·24-s + 331.·25-s − 16.2·26-s + 27·27-s − 90.9·28-s + 224.·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.91·5-s − 0.408·6-s − 1.22·7-s − 0.353·8-s + 0.333·9-s − 1.35·10-s + 1.50·11-s + 0.288·12-s + 0.172·13-s + 0.868·14-s + 1.10·15-s + 0.250·16-s + 1.03·17-s − 0.235·18-s + 0.955·20-s − 0.708·21-s − 1.06·22-s − 0.248·23-s − 0.204·24-s + 2.65·25-s − 0.122·26-s + 0.192·27-s − 0.613·28-s + 1.44·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.358712370\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.358712370\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 3 | \( 1 - 3T \) |
| 19 | \( 1 \) |
| good | 5 | \( 1 - 21.3T + 125T^{2} \) |
| 7 | \( 1 + 22.7T + 343T^{2} \) |
| 11 | \( 1 - 54.7T + 1.33e3T^{2} \) |
| 13 | \( 1 - 8.10T + 2.19e3T^{2} \) |
| 17 | \( 1 - 72.7T + 4.91e3T^{2} \) |
| 23 | \( 1 + 27.3T + 1.21e4T^{2} \) |
| 29 | \( 1 - 224.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 305.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 165.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 371.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 222.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 541.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 452.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 341.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 254.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 251.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 353.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 481.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 583.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 523.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 412.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.39e3T + 9.12e5T^{2} \) |
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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
−8.901695475542930537967913926950, −8.263930751994015957708677127117, −6.92632442676811522889070157955, −6.39652339728826062792718207563, −6.03402463518103023936250962771, −4.77303115804363250798731202363, −3.36796895533978904688034319744, −2.77138405209028466936310513368, −1.66054860614316583459314225871, −0.978742042887023599296780473539,
0.978742042887023599296780473539, 1.66054860614316583459314225871, 2.77138405209028466936310513368, 3.36796895533978904688034319744, 4.77303115804363250798731202363, 6.03402463518103023936250962771, 6.39652339728826062792718207563, 6.92632442676811522889070157955, 8.263930751994015957708677127117, 8.901695475542930537967913926950
