| L(s) = 1 | − 3-s − 3.73·5-s + 7-s + 9-s + 5·11-s + 3.19·13-s + 3.73·15-s + 17-s + 7.19·19-s − 21-s + 3·23-s + 8.92·25-s − 27-s + 4·29-s + 4.73·31-s − 5·33-s − 3.73·35-s + 4.73·37-s − 3.19·39-s − 11.7·41-s − 9.39·43-s − 3.73·45-s − 4.73·47-s + 49-s − 51-s − 5.66·53-s − 18.6·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.66·5-s + 0.377·7-s + 0.333·9-s + 1.50·11-s + 0.886·13-s + 0.963·15-s + 0.242·17-s + 1.65·19-s − 0.218·21-s + 0.625·23-s + 1.78·25-s − 0.192·27-s + 0.742·29-s + 0.849·31-s − 0.870·33-s − 0.630·35-s + 0.777·37-s − 0.511·39-s − 1.83·41-s − 1.43·43-s − 0.556·45-s − 0.690·47-s + 0.142·49-s − 0.140·51-s − 0.777·53-s − 2.51·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.531774077\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.531774077\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 + 3.73T + 5T^{2} \) |
| 11 | \( 1 - 5T + 11T^{2} \) |
| 13 | \( 1 - 3.19T + 13T^{2} \) |
| 19 | \( 1 - 7.19T + 19T^{2} \) |
| 23 | \( 1 - 3T + 23T^{2} \) |
| 29 | \( 1 - 4T + 29T^{2} \) |
| 31 | \( 1 - 4.73T + 31T^{2} \) |
| 37 | \( 1 - 4.73T + 37T^{2} \) |
| 41 | \( 1 + 11.7T + 41T^{2} \) |
| 43 | \( 1 + 9.39T + 43T^{2} \) |
| 47 | \( 1 + 4.73T + 47T^{2} \) |
| 53 | \( 1 + 5.66T + 53T^{2} \) |
| 59 | \( 1 + 2.19T + 59T^{2} \) |
| 61 | \( 1 + 6.73T + 61T^{2} \) |
| 67 | \( 1 - 11.4T + 67T^{2} \) |
| 71 | \( 1 - 3.46T + 71T^{2} \) |
| 73 | \( 1 + 3.46T + 73T^{2} \) |
| 79 | \( 1 - 0.196T + 79T^{2} \) |
| 83 | \( 1 - 12.9T + 83T^{2} \) |
| 89 | \( 1 + 12.3T + 89T^{2} \) |
| 97 | \( 1 - 13.8T + 97T^{2} \) |
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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
−8.166454627089986504467038836104, −7.36294260169292270040878392827, −6.76022319256883022210803061326, −6.14207935237932834531708920944, −4.97041840851185352614315505011, −4.57745338116478962608129149047, −3.52901142143481329884445434404, −3.31857890846164543260823749352, −1.43800941721517619041558483966, −0.77289563369327962261296290212,
0.77289563369327962261296290212, 1.43800941721517619041558483966, 3.31857890846164543260823749352, 3.52901142143481329884445434404, 4.57745338116478962608129149047, 4.97041840851185352614315505011, 6.14207935237932834531708920944, 6.76022319256883022210803061326, 7.36294260169292270040878392827, 8.166454627089986504467038836104
