| L(s) = 1 | + 3-s + 5-s − 1.82·7-s + 9-s + 1.41·11-s + 0.585·13-s + 15-s − 2.41·17-s + 7.41·19-s − 1.82·21-s + 23-s + 25-s + 27-s − 3.58·29-s + 31-s + 1.41·33-s − 1.82·35-s − 7.48·37-s + 0.585·39-s + 7.24·41-s + 7.65·43-s + 45-s + 6.24·47-s − 3.65·49-s − 2.41·51-s + 5.24·53-s + 1.41·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 0.691·7-s + 0.333·9-s + 0.426·11-s + 0.162·13-s + 0.258·15-s − 0.585·17-s + 1.70·19-s − 0.398·21-s + 0.208·23-s + 0.200·25-s + 0.192·27-s − 0.665·29-s + 0.179·31-s + 0.246·33-s − 0.309·35-s − 1.23·37-s + 0.0938·39-s + 1.13·41-s + 1.16·43-s + 0.149·45-s + 0.910·47-s − 0.522·49-s − 0.338·51-s + 0.720·53-s + 0.190·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.683317285\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.683317285\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 7 | \( 1 + 1.82T + 7T^{2} \) |
| 11 | \( 1 - 1.41T + 11T^{2} \) |
| 13 | \( 1 - 0.585T + 13T^{2} \) |
| 17 | \( 1 + 2.41T + 17T^{2} \) |
| 19 | \( 1 - 7.41T + 19T^{2} \) |
| 29 | \( 1 + 3.58T + 29T^{2} \) |
| 31 | \( 1 - T + 31T^{2} \) |
| 37 | \( 1 + 7.48T + 37T^{2} \) |
| 41 | \( 1 - 7.24T + 41T^{2} \) |
| 43 | \( 1 - 7.65T + 43T^{2} \) |
| 47 | \( 1 - 6.24T + 47T^{2} \) |
| 53 | \( 1 - 5.24T + 53T^{2} \) |
| 59 | \( 1 + 13.7T + 59T^{2} \) |
| 61 | \( 1 - 1.41T + 61T^{2} \) |
| 67 | \( 1 - 15.4T + 67T^{2} \) |
| 71 | \( 1 - 14.0T + 71T^{2} \) |
| 73 | \( 1 - 2.24T + 73T^{2} \) |
| 79 | \( 1 + 16.9T + 79T^{2} \) |
| 83 | \( 1 + 2.75T + 83T^{2} \) |
| 89 | \( 1 + 9.17T + 89T^{2} \) |
| 97 | \( 1 - 18.1T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.200634669438440961362893289719, −7.32305860495308234001708396771, −6.88612180177375309388738360845, −5.98624564788377199296581047810, −5.37403304571273071400979093884, −4.36869993005737002778636514323, −3.54957268188513932688014032935, −2.88736651361336046669144505115, −1.95723560914404829761241243371, −0.871214173717780393700615575167,
0.871214173717780393700615575167, 1.95723560914404829761241243371, 2.88736651361336046669144505115, 3.54957268188513932688014032935, 4.36869993005737002778636514323, 5.37403304571273071400979093884, 5.98624564788377199296581047810, 6.88612180177375309388738360845, 7.32305860495308234001708396771, 8.200634669438440961362893289719
