| L(s) = 1 | + 1.21·2-s − 3-s − 0.528·4-s − 5-s − 1.21·6-s − 7-s − 3.06·8-s + 9-s − 1.21·10-s − 2.26·11-s + 0.528·12-s + 4.60·13-s − 1.21·14-s + 15-s − 2.66·16-s − 4.56·17-s + 1.21·18-s − 1.01·19-s + 0.528·20-s + 21-s − 2.75·22-s + 23-s + 3.06·24-s + 25-s + 5.58·26-s − 27-s + 0.528·28-s + ⋯ |
| L(s) = 1 | + 0.857·2-s − 0.577·3-s − 0.264·4-s − 0.447·5-s − 0.495·6-s − 0.377·7-s − 1.08·8-s + 0.333·9-s − 0.383·10-s − 0.683·11-s + 0.152·12-s + 1.27·13-s − 0.324·14-s + 0.258·15-s − 0.665·16-s − 1.10·17-s + 0.285·18-s − 0.233·19-s + 0.118·20-s + 0.218·21-s − 0.586·22-s + 0.208·23-s + 0.626·24-s + 0.200·25-s + 1.09·26-s − 0.192·27-s + 0.0998·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2415 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2415 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.273828382\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.273828382\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 - 1.21T + 2T^{2} \) |
| 11 | \( 1 + 2.26T + 11T^{2} \) |
| 13 | \( 1 - 4.60T + 13T^{2} \) |
| 17 | \( 1 + 4.56T + 17T^{2} \) |
| 19 | \( 1 + 1.01T + 19T^{2} \) |
| 29 | \( 1 - 3.53T + 29T^{2} \) |
| 31 | \( 1 + 8.74T + 31T^{2} \) |
| 37 | \( 1 - 5.66T + 37T^{2} \) |
| 41 | \( 1 - 12.3T + 41T^{2} \) |
| 43 | \( 1 + 10.5T + 43T^{2} \) |
| 47 | \( 1 - 8.42T + 47T^{2} \) |
| 53 | \( 1 - 3.82T + 53T^{2} \) |
| 59 | \( 1 - 4.30T + 59T^{2} \) |
| 61 | \( 1 - 5.43T + 61T^{2} \) |
| 67 | \( 1 + 3.75T + 67T^{2} \) |
| 71 | \( 1 - 12.3T + 71T^{2} \) |
| 73 | \( 1 - 8.44T + 73T^{2} \) |
| 79 | \( 1 + 5.96T + 79T^{2} \) |
| 83 | \( 1 - 6.53T + 83T^{2} \) |
| 89 | \( 1 + 2.43T + 89T^{2} \) |
| 97 | \( 1 - 13.3T + 97T^{2} \) |
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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
−8.897865237704102515623681035263, −8.276395109799638342625086516561, −7.20645053704184466731136721824, −6.36765652322205466208783967422, −5.77699880535482360761583585678, −4.95984390883212149531571761690, −4.15037372500162828368787838593, −3.54352421118107154155658767173, −2.41130578451561721222589735356, −0.63767837520693802226342322899,
0.63767837520693802226342322899, 2.41130578451561721222589735356, 3.54352421118107154155658767173, 4.15037372500162828368787838593, 4.95984390883212149531571761690, 5.77699880535482360761583585678, 6.36765652322205466208783967422, 7.20645053704184466731136721824, 8.276395109799638342625086516561, 8.897865237704102515623681035263
