| L(s) = 1 | + 2-s + 4-s + 7-s + 8-s − 13-s + 14-s + 16-s − 7·19-s − 6·23-s − 5·25-s − 26-s + 28-s − 5·31-s + 32-s − 5·37-s − 7·38-s − 12·41-s + 11·43-s − 6·46-s + 6·47-s − 6·49-s − 5·50-s − 52-s + 12·53-s + 56-s − 6·59-s − 5·61-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.377·7-s + 0.353·8-s − 0.277·13-s + 0.267·14-s + 1/4·16-s − 1.60·19-s − 1.25·23-s − 25-s − 0.196·26-s + 0.188·28-s − 0.898·31-s + 0.176·32-s − 0.821·37-s − 1.13·38-s − 1.87·41-s + 1.67·43-s − 0.884·46-s + 0.875·47-s − 6/7·49-s − 0.707·50-s − 0.138·52-s + 1.64·53-s + 0.133·56-s − 0.781·59-s − 0.640·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5202 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5202 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 17 | \( 1 \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−7.68902014273125732423353665474, −7.15583436085154811806960522515, −6.16702040560434594459761152095, −5.77734637112887029021755678205, −4.79841488210778229056615619830, −4.17648049120606166997723849202, −3.47761270562163795419441173557, −2.30243597616580886301952977644, −1.73492689667730162764863744377, 0,
1.73492689667730162764863744377, 2.30243597616580886301952977644, 3.47761270562163795419441173557, 4.17648049120606166997723849202, 4.79841488210778229056615619830, 5.77734637112887029021755678205, 6.16702040560434594459761152095, 7.15583436085154811806960522515, 7.68902014273125732423353665474
