| L(s) = 1 | + 1.93·2-s − 0.445·3-s + 2.74·4-s − 0.192·5-s − 0.860·6-s + 1.60·7-s + 3.37·8-s − 0.801·9-s − 0.371·10-s − 1.22·12-s − 1.99·13-s + 3.10·14-s + 0.0854·15-s + 3.77·16-s − 1.67·17-s − 1.55·18-s − 1.14·19-s − 0.526·20-s − 0.713·21-s − 1.50·24-s − 0.963·25-s − 3.86·26-s + 0.801·27-s + 4.39·28-s + 0.165·30-s + 1.85·31-s + 3.94·32-s + ⋯ |
| L(s) = 1 | + 1.93·2-s − 0.445·3-s + 2.74·4-s − 0.192·5-s − 0.860·6-s + 1.60·7-s + 3.37·8-s − 0.801·9-s − 0.371·10-s − 1.22·12-s − 1.99·13-s + 3.10·14-s + 0.0854·15-s + 3.77·16-s − 1.67·17-s − 1.55·18-s − 1.14·19-s − 0.526·20-s − 0.713·21-s − 1.50·24-s − 0.963·25-s − 3.86·26-s + 0.801·27-s + 4.39·28-s + 0.165·30-s + 1.85·31-s + 3.94·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1511 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1511 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(3.006229002\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.006229002\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 1511 | \( 1 - T \) |
| good | 2 | \( 1 - 1.93T + T^{2} \) |
| 3 | \( 1 + 0.445T + T^{2} \) |
| 5 | \( 1 + 0.192T + T^{2} \) |
| 7 | \( 1 - 1.60T + T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 + 1.99T + T^{2} \) |
| 17 | \( 1 + 1.67T + T^{2} \) |
| 19 | \( 1 + 1.14T + T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - 1.85T + T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - 1.43T + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + 0.925T + T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−10.12639477196988037028153477423, −8.533731347483655720427809519220, −7.79089264388356528537900779641, −6.92435066507861896689100937975, −6.17719943329890010630000811936, −5.20101511770868752900876350705, −4.70534299018737235122005191940, −4.18574268690595340248613721297, −2.58147469280042499496751183195, −2.10749286607126491516318985264,
2.10749286607126491516318985264, 2.58147469280042499496751183195, 4.18574268690595340248613721297, 4.70534299018737235122005191940, 5.20101511770868752900876350705, 6.17719943329890010630000811936, 6.92435066507861896689100937975, 7.79089264388356528537900779641, 8.533731347483655720427809519220, 10.12639477196988037028153477423
