| L(s) = 1 | + 1.19·2-s − 2.27·3-s − 0.574·4-s − 5-s − 2.71·6-s − 2.19·7-s − 3.07·8-s + 2.16·9-s − 1.19·10-s − 2.82·11-s + 1.30·12-s − 4.08·13-s − 2.62·14-s + 2.27·15-s − 2.52·16-s + 0.382·17-s + 2.58·18-s + 0.574·20-s + 4.99·21-s − 3.37·22-s − 2.97·23-s + 6.98·24-s + 25-s − 4.87·26-s + 1.89·27-s + 1.26·28-s − 8.57·29-s + ⋯ |
| L(s) = 1 | + 0.844·2-s − 1.31·3-s − 0.287·4-s − 0.447·5-s − 1.10·6-s − 0.830·7-s − 1.08·8-s + 0.722·9-s − 0.377·10-s − 0.852·11-s + 0.376·12-s − 1.13·13-s − 0.700·14-s + 0.586·15-s − 0.630·16-s + 0.0928·17-s + 0.609·18-s + 0.128·20-s + 1.08·21-s − 0.719·22-s − 0.620·23-s + 1.42·24-s + 0.200·25-s − 0.955·26-s + 0.364·27-s + 0.238·28-s − 1.59·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3822880768\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3822880768\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 - 1.19T + 2T^{2} \) |
| 3 | \( 1 + 2.27T + 3T^{2} \) |
| 7 | \( 1 + 2.19T + 7T^{2} \) |
| 11 | \( 1 + 2.82T + 11T^{2} \) |
| 13 | \( 1 + 4.08T + 13T^{2} \) |
| 17 | \( 1 - 0.382T + 17T^{2} \) |
| 23 | \( 1 + 2.97T + 23T^{2} \) |
| 29 | \( 1 + 8.57T + 29T^{2} \) |
| 31 | \( 1 - 4.69T + 31T^{2} \) |
| 37 | \( 1 - 10.7T + 37T^{2} \) |
| 41 | \( 1 + 3.13T + 41T^{2} \) |
| 43 | \( 1 - 2.00T + 43T^{2} \) |
| 47 | \( 1 + 10.8T + 47T^{2} \) |
| 53 | \( 1 + 7.47T + 53T^{2} \) |
| 59 | \( 1 - 12.9T + 59T^{2} \) |
| 61 | \( 1 - 5.97T + 61T^{2} \) |
| 67 | \( 1 + 1.04T + 67T^{2} \) |
| 71 | \( 1 - 7.82T + 71T^{2} \) |
| 73 | \( 1 + 14.0T + 73T^{2} \) |
| 79 | \( 1 - 13.4T + 79T^{2} \) |
| 83 | \( 1 + 5.55T + 83T^{2} \) |
| 89 | \( 1 - 6.56T + 89T^{2} \) |
| 97 | \( 1 + 14.4T + 97T^{2} \) |
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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
−9.670622103789851932940179411867, −8.381166724430604376593401727541, −7.50264365570427681920670325372, −6.53157883946383708981955606420, −5.90472418540255798282314370675, −5.14888303432923983272769079149, −4.58544207086304952381579006513, −3.57148284426678461526587329762, −2.58383508635090023943248208434, −0.37158441969459668925570043229,
0.37158441969459668925570043229, 2.58383508635090023943248208434, 3.57148284426678461526587329762, 4.58544207086304952381579006513, 5.14888303432923983272769079149, 5.90472418540255798282314370675, 6.53157883946383708981955606420, 7.50264365570427681920670325372, 8.381166724430604376593401727541, 9.670622103789851932940179411867
