| L(s) = 1 | + 3-s + 4·7-s + 9-s − 2·11-s − 13-s + 2·19-s + 4·21-s − 2·23-s + 27-s − 10·29-s − 4·31-s − 2·33-s + 6·37-s − 39-s − 6·41-s − 8·43-s + 12·47-s + 9·49-s + 14·53-s + 2·57-s + 6·59-s − 2·61-s + 4·63-s + 4·67-s − 2·69-s + 14·73-s − 8·77-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.51·7-s + 1/3·9-s − 0.603·11-s − 0.277·13-s + 0.458·19-s + 0.872·21-s − 0.417·23-s + 0.192·27-s − 1.85·29-s − 0.718·31-s − 0.348·33-s + 0.986·37-s − 0.160·39-s − 0.937·41-s − 1.21·43-s + 1.75·47-s + 9/7·49-s + 1.92·53-s + 0.264·57-s + 0.781·59-s − 0.256·61-s + 0.503·63-s + 0.488·67-s − 0.240·69-s + 1.63·73-s − 0.911·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.517491509\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.517491509\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−14.39715484103410, −13.66893038516462, −13.44554475980138, −12.88231466398937, −12.11137479257000, −11.82372433020379, −11.10560104618083, −10.86902583041266, −10.12823864330449, −9.712339619387235, −8.967928123103721, −8.642186913933681, −7.953256095238052, −7.601532779414468, −7.268020674188350, −6.453607835499414, −5.548004621765014, −5.276070001857497, −4.728882114890786, −3.879786002035091, −3.621609083321018, −2.504975730840971, −2.154237056822741, −1.507697832670573, −0.6073811201347955,
0.6073811201347955, 1.507697832670573, 2.154237056822741, 2.504975730840971, 3.621609083321018, 3.879786002035091, 4.728882114890786, 5.276070001857497, 5.548004621765014, 6.453607835499414, 7.268020674188350, 7.601532779414468, 7.953256095238052, 8.642186913933681, 8.967928123103721, 9.712339619387235, 10.12823864330449, 10.86902583041266, 11.10560104618083, 11.82372433020379, 12.11137479257000, 12.88231466398937, 13.44554475980138, 13.66893038516462, 14.39715484103410
