| L(s) = 1 | + 2.29·3-s + 0.952·5-s + 7-s + 2.28·9-s + 11-s − 13-s + 2.18·15-s + 3.49·17-s + 2.81·19-s + 2.29·21-s − 5.65·23-s − 4.09·25-s − 1.65·27-s + 2.02·29-s + 6.06·31-s + 2.29·33-s + 0.952·35-s + 11.7·37-s − 2.29·39-s + 1.38·41-s + 1.18·43-s + 2.17·45-s + 10.8·47-s + 49-s + 8.03·51-s + 9.25·53-s + 0.952·55-s + ⋯ |
| L(s) = 1 | + 1.32·3-s + 0.425·5-s + 0.377·7-s + 0.760·9-s + 0.301·11-s − 0.277·13-s + 0.565·15-s + 0.847·17-s + 0.646·19-s + 0.501·21-s − 1.17·23-s − 0.818·25-s − 0.317·27-s + 0.375·29-s + 1.08·31-s + 0.400·33-s + 0.160·35-s + 1.92·37-s − 0.367·39-s + 0.216·41-s + 0.180·43-s + 0.323·45-s + 1.58·47-s + 0.142·49-s + 1.12·51-s + 1.27·53-s + 0.128·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.734614769\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.734614769\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 - 2.29T + 3T^{2} \) |
| 5 | \( 1 - 0.952T + 5T^{2} \) |
| 17 | \( 1 - 3.49T + 17T^{2} \) |
| 19 | \( 1 - 2.81T + 19T^{2} \) |
| 23 | \( 1 + 5.65T + 23T^{2} \) |
| 29 | \( 1 - 2.02T + 29T^{2} \) |
| 31 | \( 1 - 6.06T + 31T^{2} \) |
| 37 | \( 1 - 11.7T + 37T^{2} \) |
| 41 | \( 1 - 1.38T + 41T^{2} \) |
| 43 | \( 1 - 1.18T + 43T^{2} \) |
| 47 | \( 1 - 10.8T + 47T^{2} \) |
| 53 | \( 1 - 9.25T + 53T^{2} \) |
| 59 | \( 1 + 5.22T + 59T^{2} \) |
| 61 | \( 1 + 9.26T + 61T^{2} \) |
| 67 | \( 1 - 9.82T + 67T^{2} \) |
| 71 | \( 1 + 5.77T + 71T^{2} \) |
| 73 | \( 1 - 10.7T + 73T^{2} \) |
| 79 | \( 1 - 1.51T + 79T^{2} \) |
| 83 | \( 1 + 4.09T + 83T^{2} \) |
| 89 | \( 1 + 12.8T + 89T^{2} \) |
| 97 | \( 1 + 2.00T + 97T^{2} \) |
| 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
−8.335667457118358185238234554144, −7.85777485903286956968691247366, −7.30744762700309293170569710586, −6.15235729625760836683615372157, −5.58376917196897597297387255823, −4.43638748725279041081577002243, −3.78741849396660142630946572452, −2.79816978116135269120008604209, −2.20260720843351760001205810270, −1.11084746127321322104108706742,
1.11084746127321322104108706742, 2.20260720843351760001205810270, 2.79816978116135269120008604209, 3.78741849396660142630946572452, 4.43638748725279041081577002243, 5.58376917196897597297387255823, 6.15235729625760836683615372157, 7.30744762700309293170569710586, 7.85777485903286956968691247366, 8.335667457118358185238234554144
