| L(s) = 1 | + 0.465·3-s + 3.21·5-s − 7-s − 2.78·9-s − 1.43·11-s + 4.35·13-s + 1.49·15-s + 3.88·17-s + 4.37·19-s − 0.465·21-s − 23-s + 5.36·25-s − 2.68·27-s − 7.29·29-s − 1.13·31-s − 0.666·33-s − 3.21·35-s + 11.3·37-s + 2.02·39-s + 0.579·41-s + 12.0·43-s − 8.96·45-s − 8.87·47-s + 49-s + 1.80·51-s + 14.0·53-s − 4.61·55-s + ⋯ |
| L(s) = 1 | + 0.268·3-s + 1.43·5-s − 0.377·7-s − 0.927·9-s − 0.432·11-s + 1.20·13-s + 0.386·15-s + 0.942·17-s + 1.00·19-s − 0.101·21-s − 0.208·23-s + 1.07·25-s − 0.517·27-s − 1.35·29-s − 0.203·31-s − 0.116·33-s − 0.544·35-s + 1.85·37-s + 0.324·39-s + 0.0904·41-s + 1.83·43-s − 1.33·45-s − 1.29·47-s + 0.142·49-s + 0.253·51-s + 1.93·53-s − 0.622·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.543058765\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.543058765\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 - 0.465T + 3T^{2} \) |
| 5 | \( 1 - 3.21T + 5T^{2} \) |
| 11 | \( 1 + 1.43T + 11T^{2} \) |
| 13 | \( 1 - 4.35T + 13T^{2} \) |
| 17 | \( 1 - 3.88T + 17T^{2} \) |
| 19 | \( 1 - 4.37T + 19T^{2} \) |
| 29 | \( 1 + 7.29T + 29T^{2} \) |
| 31 | \( 1 + 1.13T + 31T^{2} \) |
| 37 | \( 1 - 11.3T + 37T^{2} \) |
| 41 | \( 1 - 0.579T + 41T^{2} \) |
| 43 | \( 1 - 12.0T + 43T^{2} \) |
| 47 | \( 1 + 8.87T + 47T^{2} \) |
| 53 | \( 1 - 14.0T + 53T^{2} \) |
| 59 | \( 1 + 2.48T + 59T^{2} \) |
| 61 | \( 1 - 2.08T + 61T^{2} \) |
| 67 | \( 1 - 7.64T + 67T^{2} \) |
| 71 | \( 1 - 5.42T + 71T^{2} \) |
| 73 | \( 1 - 0.982T + 73T^{2} \) |
| 79 | \( 1 + 13.1T + 79T^{2} \) |
| 83 | \( 1 - 12.1T + 83T^{2} \) |
| 89 | \( 1 + 12.6T + 89T^{2} \) |
| 97 | \( 1 + 9.22T + 97T^{2} \) |
| 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
−9.081188285542768704408145164296, −8.177044143372854240113693791331, −7.45533997368661079454220253846, −6.32056830889486699350029274381, −5.69883401039667939472898676837, −5.41548740492753653534938456712, −3.89863837277495091593123526702, −3.02320149100444020687918327686, −2.23036777245505926292132973170, −1.03998814480201573097307210199,
1.03998814480201573097307210199, 2.23036777245505926292132973170, 3.02320149100444020687918327686, 3.89863837277495091593123526702, 5.41548740492753653534938456712, 5.69883401039667939472898676837, 6.32056830889486699350029274381, 7.45533997368661079454220253846, 8.177044143372854240113693791331, 9.081188285542768704408145164296
