L(s) = 1 | + 2-s + 4-s + 8-s + 11-s − 13-s + 16-s + 6·17-s + 2·19-s + 22-s + 6·23-s − 5·25-s − 26-s − 9·29-s − 4·31-s + 32-s + 6·34-s + 2·37-s + 2·38-s + 6·41-s − 4·43-s + 44-s + 6·46-s + 6·47-s − 5·50-s − 52-s − 9·58-s + 3·59-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s + 0.301·11-s − 0.277·13-s + 1/4·16-s + 1.45·17-s + 0.458·19-s + 0.213·22-s + 1.25·23-s − 25-s − 0.196·26-s − 1.67·29-s − 0.718·31-s + 0.176·32-s + 1.02·34-s + 0.328·37-s + 0.324·38-s + 0.937·41-s − 0.609·43-s + 0.150·44-s + 0.884·46-s + 0.875·47-s − 0.707·50-s − 0.138·52-s − 1.18·58-s + 0.390·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.685523905\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.685523905\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
good | 5 | \( 1 + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 - 11 T + p T^{2} \) |
| 67 | \( 1 - 11 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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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
−7.56704530776783020057178969772, −7.02595354834647276315365345721, −6.20497672746778586371330345474, −5.34528774307920452062194632613, −5.25186180970575653848515017825, −3.97278631808423878087281306386, −3.63097767864186403022019852192, −2.73086151356196192613985265169, −1.84018665495927734624917711462, −0.839928885383839087156714802594,
0.839928885383839087156714802594, 1.84018665495927734624917711462, 2.73086151356196192613985265169, 3.63097767864186403022019852192, 3.97278631808423878087281306386, 5.25186180970575653848515017825, 5.34528774307920452062194632613, 6.20497672746778586371330345474, 7.02595354834647276315365345721, 7.56704530776783020057178969772