| L(s) = 1 | − 1.21·2-s + 3-s − 0.525·4-s − 1.21·6-s + 4.90·7-s + 3.06·8-s + 9-s − 0.525·12-s + 4.14·13-s − 5.95·14-s − 2.67·16-s − 5.33·17-s − 1.21·18-s + 5.18·19-s + 4.90·21-s − 4·23-s + 3.06·24-s − 5.03·26-s + 27-s − 2.57·28-s − 1.80·29-s + 2.62·31-s − 2.88·32-s + 6.47·34-s − 0.525·36-s − 5.80·37-s − 6.29·38-s + ⋯ |
| L(s) = 1 | − 0.858·2-s + 0.577·3-s − 0.262·4-s − 0.495·6-s + 1.85·7-s + 1.08·8-s + 0.333·9-s − 0.151·12-s + 1.15·13-s − 1.59·14-s − 0.668·16-s − 1.29·17-s − 0.286·18-s + 1.18·19-s + 1.06·21-s − 0.834·23-s + 0.625·24-s − 0.987·26-s + 0.192·27-s − 0.486·28-s − 0.335·29-s + 0.470·31-s − 0.510·32-s + 1.11·34-s − 0.0875·36-s − 0.954·37-s − 1.02·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.051482580\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.051482580\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 1.21T + 2T^{2} \) |
| 7 | \( 1 - 4.90T + 7T^{2} \) |
| 13 | \( 1 - 4.14T + 13T^{2} \) |
| 17 | \( 1 + 5.33T + 17T^{2} \) |
| 19 | \( 1 - 5.18T + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 + 1.80T + 29T^{2} \) |
| 31 | \( 1 - 2.62T + 31T^{2} \) |
| 37 | \( 1 + 5.80T + 37T^{2} \) |
| 41 | \( 1 + 1.80T + 41T^{2} \) |
| 43 | \( 1 - 4.90T + 43T^{2} \) |
| 47 | \( 1 + 7.05T + 47T^{2} \) |
| 53 | \( 1 - 7.18T + 53T^{2} \) |
| 59 | \( 1 - 1.67T + 59T^{2} \) |
| 61 | \( 1 + 0.755T + 61T^{2} \) |
| 67 | \( 1 - 4.85T + 67T^{2} \) |
| 71 | \( 1 - 0.428T + 71T^{2} \) |
| 73 | \( 1 - 12.7T + 73T^{2} \) |
| 79 | \( 1 - 6.42T + 79T^{2} \) |
| 83 | \( 1 + 2.90T + 83T^{2} \) |
| 89 | \( 1 - 0.622T + 89T^{2} \) |
| 97 | \( 1 - 2.75T + 97T^{2} \) |
| 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
−7.956000158192831750543279215379, −7.42310383692604317263442017640, −6.61325391178579816190892295001, −5.51720600073119788253420642123, −4.88869924655958038353473046937, −4.22852520169071384707905396936, −3.58088400067104101410860796764, −2.21355586031351909025284776293, −1.64638984761595203001268343530, −0.833837575694138690640980516308,
0.833837575694138690640980516308, 1.64638984761595203001268343530, 2.21355586031351909025284776293, 3.58088400067104101410860796764, 4.22852520169071384707905396936, 4.88869924655958038353473046937, 5.51720600073119788253420642123, 6.61325391178579816190892295001, 7.42310383692604317263442017640, 7.956000158192831750543279215379
