| L(s) = 1 | + 17·7-s − 89·13-s − 107·19-s − 125·25-s + 308·31-s + 433·37-s + 520·43-s − 54·49-s + 901·61-s − 1.00e3·67-s − 271·73-s + 503·79-s − 1.51e3·91-s + 1.85e3·97-s − 19·103-s + 646·109-s + ⋯ |
| L(s) = 1 | + 0.917·7-s − 1.89·13-s − 1.29·19-s − 25-s + 1.78·31-s + 1.92·37-s + 1.84·43-s − 0.157·49-s + 1.89·61-s − 1.83·67-s − 0.434·73-s + 0.716·79-s − 1.74·91-s + 1.93·97-s − 0.0181·103-s + 0.567·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.919323602\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.919323602\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + p^{3} T^{2} \) |
| 7 | \( 1 - 17 T + p^{3} T^{2} \) |
| 11 | \( 1 + p^{3} T^{2} \) |
| 13 | \( 1 + 89 T + p^{3} T^{2} \) |
| 17 | \( 1 + p^{3} T^{2} \) |
| 19 | \( 1 + 107 T + p^{3} T^{2} \) |
| 23 | \( 1 + p^{3} T^{2} \) |
| 29 | \( 1 + p^{3} T^{2} \) |
| 31 | \( 1 - 308 T + p^{3} T^{2} \) |
| 37 | \( 1 - 433 T + p^{3} T^{2} \) |
| 41 | \( 1 + p^{3} T^{2} \) |
| 43 | \( 1 - 520 T + p^{3} T^{2} \) |
| 47 | \( 1 + p^{3} T^{2} \) |
| 53 | \( 1 + p^{3} T^{2} \) |
| 59 | \( 1 + p^{3} T^{2} \) |
| 61 | \( 1 - 901 T + p^{3} T^{2} \) |
| 67 | \( 1 + 1007 T + p^{3} T^{2} \) |
| 71 | \( 1 + p^{3} T^{2} \) |
| 73 | \( 1 + 271 T + p^{3} T^{2} \) |
| 79 | \( 1 - 503 T + p^{3} T^{2} \) |
| 83 | \( 1 + p^{3} T^{2} \) |
| 89 | \( 1 + p^{3} T^{2} \) |
| 97 | \( 1 - 1853 T + p^{3} T^{2} \) |
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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
−8.955669921499257343960176174911, −7.968790456410313402300298963552, −7.61652506631906012275308334162, −6.56545592702989265554532401631, −5.69567164603722165861199032355, −4.64990097885455793609559870758, −4.28180537689561779652291402681, −2.69126841543013924088549871818, −2.05012142655228164502731962092, −0.64169872504736194092209789466,
0.64169872504736194092209789466, 2.05012142655228164502731962092, 2.69126841543013924088549871818, 4.28180537689561779652291402681, 4.64990097885455793609559870758, 5.69567164603722165861199032355, 6.56545592702989265554532401631, 7.61652506631906012275308334162, 7.968790456410313402300298963552, 8.955669921499257343960176174911
