L(s) = 1 | + 1.69·3-s − 2.62·5-s − 7-s − 0.136·9-s − 11-s + 13-s − 4.44·15-s + 2.10·17-s + 4.75·19-s − 1.69·21-s + 7.30·23-s + 1.89·25-s − 5.30·27-s − 8.99·29-s + 4.35·31-s − 1.69·33-s + 2.62·35-s − 10.9·37-s + 1.69·39-s − 2.97·41-s − 6.62·43-s + 0.358·45-s + 8.67·47-s + 49-s + 3.56·51-s + 11.5·53-s + 2.62·55-s + ⋯ |
L(s) = 1 | + 0.976·3-s − 1.17·5-s − 0.377·7-s − 0.0455·9-s − 0.301·11-s + 0.277·13-s − 1.14·15-s + 0.510·17-s + 1.09·19-s − 0.369·21-s + 1.52·23-s + 0.379·25-s − 1.02·27-s − 1.67·29-s + 0.782·31-s − 0.294·33-s + 0.443·35-s − 1.80·37-s + 0.270·39-s − 0.463·41-s − 1.00·43-s + 0.0534·45-s + 1.26·47-s + 0.142·49-s + 0.498·51-s + 1.58·53-s + 0.354·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.807956538\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.807956538\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
good | 3 | \( 1 - 1.69T + 3T^{2} \) |
| 5 | \( 1 + 2.62T + 5T^{2} \) |
| 17 | \( 1 - 2.10T + 17T^{2} \) |
| 19 | \( 1 - 4.75T + 19T^{2} \) |
| 23 | \( 1 - 7.30T + 23T^{2} \) |
| 29 | \( 1 + 8.99T + 29T^{2} \) |
| 31 | \( 1 - 4.35T + 31T^{2} \) |
| 37 | \( 1 + 10.9T + 37T^{2} \) |
| 41 | \( 1 + 2.97T + 41T^{2} \) |
| 43 | \( 1 + 6.62T + 43T^{2} \) |
| 47 | \( 1 - 8.67T + 47T^{2} \) |
| 53 | \( 1 - 11.5T + 53T^{2} \) |
| 59 | \( 1 + 8.45T + 59T^{2} \) |
| 61 | \( 1 + 2.44T + 61T^{2} \) |
| 67 | \( 1 + 11.0T + 67T^{2} \) |
| 71 | \( 1 - 4.84T + 71T^{2} \) |
| 73 | \( 1 - 8.55T + 73T^{2} \) |
| 79 | \( 1 - 12.9T + 79T^{2} \) |
| 83 | \( 1 - 13.4T + 83T^{2} \) |
| 89 | \( 1 - 0.0240T + 89T^{2} \) |
| 97 | \( 1 - 3.50T + 97T^{2} \) |
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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
−7.73666319354062302839548798702, −7.43889409787218971895220324252, −6.71041999411053962451959723315, −5.60120320408238291566343777085, −5.05651920725345903224007481845, −3.96291795831526449273244432987, −3.36745940633454713227639189035, −3.03430339127301798277277114861, −1.88713200104176224223087582766, −0.62100668057755317296916496609,
0.62100668057755317296916496609, 1.88713200104176224223087582766, 3.03430339127301798277277114861, 3.36745940633454713227639189035, 3.96291795831526449273244432987, 5.05651920725345903224007481845, 5.60120320408238291566343777085, 6.71041999411053962451959723315, 7.43889409787218971895220324252, 7.73666319354062302839548798702