| L(s) = 1 | − 2.74e3·3-s + 9.76e6·5-s + 1.17e9·7-s − 1.04e10·9-s + 1.43e11·11-s − 7.93e11·13-s − 2.68e10·15-s − 1.36e13·17-s − 1.04e13·19-s − 3.21e12·21-s + 1.69e14·23-s + 9.53e13·25-s + 5.74e13·27-s + 3.64e15·29-s − 3.42e15·31-s − 3.94e14·33-s + 1.14e16·35-s + 1.89e14·37-s + 2.17e15·39-s + 2.67e16·41-s + 1.93e17·43-s − 1.02e17·45-s + 3.18e16·47-s + 8.13e17·49-s + 3.73e16·51-s + 1.37e18·53-s + 1.40e18·55-s + ⋯ |
| L(s) = 1 | − 0.0268·3-s + 0.447·5-s + 1.56·7-s − 0.999·9-s + 1.66·11-s − 1.59·13-s − 0.0120·15-s − 1.63·17-s − 0.391·19-s − 0.0420·21-s + 0.851·23-s + 0.199·25-s + 0.0536·27-s + 1.60·29-s − 0.750·31-s − 0.0447·33-s + 0.700·35-s + 0.00648·37-s + 0.0428·39-s + 0.311·41-s + 1.36·43-s − 0.446·45-s + 0.0884·47-s + 1.45·49-s + 0.0439·51-s + 1.07·53-s + 0.746·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(11)\) |
\(\approx\) |
\(2.853817850\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.853817850\) |
| \(L(\frac{23}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 9.76e6T \) |
| good | 3 | \( 1 + 2.74e3T + 1.04e10T^{2} \) |
| 7 | \( 1 - 1.17e9T + 5.58e17T^{2} \) |
| 11 | \( 1 - 1.43e11T + 7.40e21T^{2} \) |
| 13 | \( 1 + 7.93e11T + 2.47e23T^{2} \) |
| 17 | \( 1 + 1.36e13T + 6.90e25T^{2} \) |
| 19 | \( 1 + 1.04e13T + 7.14e26T^{2} \) |
| 23 | \( 1 - 1.69e14T + 3.94e28T^{2} \) |
| 29 | \( 1 - 3.64e15T + 5.13e30T^{2} \) |
| 31 | \( 1 + 3.42e15T + 2.08e31T^{2} \) |
| 37 | \( 1 - 1.89e14T + 8.55e32T^{2} \) |
| 41 | \( 1 - 2.67e16T + 7.38e33T^{2} \) |
| 43 | \( 1 - 1.93e17T + 2.00e34T^{2} \) |
| 47 | \( 1 - 3.18e16T + 1.30e35T^{2} \) |
| 53 | \( 1 - 1.37e18T + 1.62e36T^{2} \) |
| 59 | \( 1 + 1.02e18T + 1.54e37T^{2} \) |
| 61 | \( 1 + 8.67e18T + 3.10e37T^{2} \) |
| 67 | \( 1 - 6.74e17T + 2.22e38T^{2} \) |
| 71 | \( 1 - 3.43e18T + 7.52e38T^{2} \) |
| 73 | \( 1 + 2.16e19T + 1.34e39T^{2} \) |
| 79 | \( 1 - 6.59e19T + 7.08e39T^{2} \) |
| 83 | \( 1 - 1.85e20T + 1.99e40T^{2} \) |
| 89 | \( 1 + 4.78e20T + 8.65e40T^{2} \) |
| 97 | \( 1 - 7.66e19T + 5.27e41T^{2} \) |
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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
−10.73986821369585412605002784911, −9.202079912746426207741504255844, −8.639803432561938417541508856267, −7.31062992050076163710202785736, −6.26719586197965773881670030871, −4.99375760369858286337274243559, −4.32557820089285080233769860053, −2.61814630150373097909495036921, −1.83600705210246784708580831569, −0.69665729252466933950838290408,
0.69665729252466933950838290408, 1.83600705210246784708580831569, 2.61814630150373097909495036921, 4.32557820089285080233769860053, 4.99375760369858286337274243559, 6.26719586197965773881670030871, 7.31062992050076163710202785736, 8.639803432561938417541508856267, 9.202079912746426207741504255844, 10.73986821369585412605002784911
