| L(s) = 1 | − 1.18·3-s + 0.375·5-s + 4.19·7-s − 1.58·9-s + 6.14·11-s + 3.39·13-s − 0.446·15-s − 2.63·17-s + 2.89·19-s − 4.98·21-s − 1.99·23-s − 4.85·25-s + 5.45·27-s + 10.4·31-s − 7.31·33-s + 1.57·35-s + 6.80·37-s − 4.03·39-s − 4.75·41-s + 10.5·43-s − 0.595·45-s − 1.56·47-s + 10.5·49-s + 3.13·51-s − 2.41·53-s + 2.30·55-s − 3.44·57-s + ⋯ |
| L(s) = 1 | − 0.686·3-s + 0.167·5-s + 1.58·7-s − 0.528·9-s + 1.85·11-s + 0.940·13-s − 0.115·15-s − 0.639·17-s + 0.664·19-s − 1.08·21-s − 0.415·23-s − 0.971·25-s + 1.04·27-s + 1.87·31-s − 1.27·33-s + 0.266·35-s + 1.11·37-s − 0.645·39-s − 0.742·41-s + 1.60·43-s − 0.0887·45-s − 0.228·47-s + 1.51·49-s + 0.438·51-s − 0.331·53-s + 0.311·55-s − 0.456·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.424102225\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.424102225\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 29 | \( 1 \) |
| good | 3 | \( 1 + 1.18T + 3T^{2} \) |
| 5 | \( 1 - 0.375T + 5T^{2} \) |
| 7 | \( 1 - 4.19T + 7T^{2} \) |
| 11 | \( 1 - 6.14T + 11T^{2} \) |
| 13 | \( 1 - 3.39T + 13T^{2} \) |
| 17 | \( 1 + 2.63T + 17T^{2} \) |
| 19 | \( 1 - 2.89T + 19T^{2} \) |
| 23 | \( 1 + 1.99T + 23T^{2} \) |
| 31 | \( 1 - 10.4T + 31T^{2} \) |
| 37 | \( 1 - 6.80T + 37T^{2} \) |
| 41 | \( 1 + 4.75T + 41T^{2} \) |
| 43 | \( 1 - 10.5T + 43T^{2} \) |
| 47 | \( 1 + 1.56T + 47T^{2} \) |
| 53 | \( 1 + 2.41T + 53T^{2} \) |
| 59 | \( 1 - 2.09T + 59T^{2} \) |
| 61 | \( 1 - 8.35T + 61T^{2} \) |
| 67 | \( 1 - 3.49T + 67T^{2} \) |
| 71 | \( 1 + 9.34T + 71T^{2} \) |
| 73 | \( 1 + 2.46T + 73T^{2} \) |
| 79 | \( 1 + 3.01T + 79T^{2} \) |
| 83 | \( 1 + 7.61T + 83T^{2} \) |
| 89 | \( 1 + 4.71T + 89T^{2} \) |
| 97 | \( 1 - 13.3T + 97T^{2} \) |
| 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
−8.149096673938219803249745223692, −7.22610298560962718316716367448, −6.30306896744411010383620015255, −6.03001019042648540531355505446, −5.16632277863805601405823298045, −4.37370525105364029809559171529, −3.89301860415684446190682844381, −2.61920743564708191992285851777, −1.54298838418919242556043432534, −0.937367424066198642761802048799,
0.937367424066198642761802048799, 1.54298838418919242556043432534, 2.61920743564708191992285851777, 3.89301860415684446190682844381, 4.37370525105364029809559171529, 5.16632277863805601405823298045, 6.03001019042648540531355505446, 6.30306896744411010383620015255, 7.22610298560962718316716367448, 8.149096673938219803249745223692
