| L(s) = 1 | + 3·3-s + 18·5-s + 7·7-s + 9·9-s + 44·11-s − 58·13-s + 54·15-s − 130·17-s + 92·19-s + 21·21-s − 84·23-s + 199·25-s + 27·27-s + 250·29-s + 72·31-s + 132·33-s + 126·35-s + 354·37-s − 174·39-s + 334·41-s − 416·43-s + 162·45-s + 464·47-s + 49·49-s − 390·51-s + 450·53-s + 792·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.60·5-s + 0.377·7-s + 1/3·9-s + 1.20·11-s − 1.23·13-s + 0.929·15-s − 1.85·17-s + 1.11·19-s + 0.218·21-s − 0.761·23-s + 1.59·25-s + 0.192·27-s + 1.60·29-s + 0.417·31-s + 0.696·33-s + 0.608·35-s + 1.57·37-s − 0.714·39-s + 1.27·41-s − 1.47·43-s + 0.536·45-s + 1.44·47-s + 1/7·49-s − 1.07·51-s + 1.16·53-s + 1.94·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.203480687\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.203480687\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p T \) |
| 7 | \( 1 - p T \) |
| good | 5 | \( 1 - 18 T + p^{3} T^{2} \) |
| 11 | \( 1 - 4 p T + p^{3} T^{2} \) |
| 13 | \( 1 + 58 T + p^{3} T^{2} \) |
| 17 | \( 1 + 130 T + p^{3} T^{2} \) |
| 19 | \( 1 - 92 T + p^{3} T^{2} \) |
| 23 | \( 1 + 84 T + p^{3} T^{2} \) |
| 29 | \( 1 - 250 T + p^{3} T^{2} \) |
| 31 | \( 1 - 72 T + p^{3} T^{2} \) |
| 37 | \( 1 - 354 T + p^{3} T^{2} \) |
| 41 | \( 1 - 334 T + p^{3} T^{2} \) |
| 43 | \( 1 + 416 T + p^{3} T^{2} \) |
| 47 | \( 1 - 464 T + p^{3} T^{2} \) |
| 53 | \( 1 - 450 T + p^{3} T^{2} \) |
| 59 | \( 1 + 516 T + p^{3} T^{2} \) |
| 61 | \( 1 + 58 T + p^{3} T^{2} \) |
| 67 | \( 1 + 656 T + p^{3} T^{2} \) |
| 71 | \( 1 - 940 T + p^{3} T^{2} \) |
| 73 | \( 1 - 178 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1072 T + p^{3} T^{2} \) |
| 83 | \( 1 - 660 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1254 T + p^{3} T^{2} \) |
| 97 | \( 1 - 210 T + p^{3} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.305582703868373065172437325914, −8.705878216711770656081188849840, −7.59148513820323090596529032598, −6.66742518518297985822625352687, −6.08630824929118108074270219531, −4.93972727529689694707084223187, −4.24051183143255344225387649465, −2.69950950161792124348077473658, −2.13109035890668212634094405544, −1.06069406897629801847285826974,
1.06069406897629801847285826974, 2.13109035890668212634094405544, 2.69950950161792124348077473658, 4.24051183143255344225387649465, 4.93972727529689694707084223187, 6.08630824929118108074270219531, 6.66742518518297985822625352687, 7.59148513820323090596529032598, 8.705878216711770656081188849840, 9.305582703868373065172437325914
