| L(s) = 1 | + 3-s + 7-s + 9-s − 3·17-s − 4·19-s + 21-s + 3·23-s + 27-s + 10·29-s + 3·31-s − 4·37-s + 4·41-s − 6·43-s + 9·47-s + 49-s − 3·51-s − 13·53-s − 4·57-s + 14·59-s − 13·61-s + 63-s + 2·67-s + 3·69-s − 2·71-s − 4·73-s + 79-s + 81-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.377·7-s + 1/3·9-s − 0.727·17-s − 0.917·19-s + 0.218·21-s + 0.625·23-s + 0.192·27-s + 1.85·29-s + 0.538·31-s − 0.657·37-s + 0.624·41-s − 0.914·43-s + 1.31·47-s + 1/7·49-s − 0.420·51-s − 1.78·53-s − 0.529·57-s + 1.82·59-s − 1.66·61-s + 0.125·63-s + 0.244·67-s + 0.361·69-s − 0.237·71-s − 0.468·73-s + 0.112·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 254100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 254100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.300085206\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.300085206\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 + 13 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−12.87576894009675, −12.35257496230544, −12.05709611911078, −11.38927629669042, −10.92479263546755, −10.58690388558508, −10.00429572160037, −9.661120545657111, −8.856073208241182, −8.688173972471229, −8.314635831791144, −7.714319098411426, −7.188867779429422, −6.706405235933972, −6.285112462174607, −5.731718034278875, −4.922945364864765, −4.604780939121274, −4.220290643617015, −3.464043198067824, −2.935634825182886, −2.423165335943862, −1.866693274643024, −1.204271123760299, −0.4874487406980892,
0.4874487406980892, 1.204271123760299, 1.866693274643024, 2.423165335943862, 2.935634825182886, 3.464043198067824, 4.220290643617015, 4.604780939121274, 4.922945364864765, 5.731718034278875, 6.285112462174607, 6.706405235933972, 7.188867779429422, 7.714319098411426, 8.314635831791144, 8.688173972471229, 8.856073208241182, 9.661120545657111, 10.00429572160037, 10.58690388558508, 10.92479263546755, 11.38927629669042, 12.05709611911078, 12.35257496230544, 12.87576894009675
