| L(s) = 1 | + 2·3-s + 6·7-s + 2·9-s + 12·21-s + 2·23-s + 6·27-s − 12·29-s + 18·43-s + 14·47-s + 18·49-s + 12·63-s + 6·67-s + 4·69-s + 11·81-s − 22·83-s − 24·87-s + 12·89-s − 36·101-s + 18·103-s − 26·107-s − 22·121-s + 127-s + 36·129-s + 131-s + 137-s + 139-s + 28·141-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 2.26·7-s + 2/3·9-s + 2.61·21-s + 0.417·23-s + 1.15·27-s − 2.22·29-s + 2.74·43-s + 2.04·47-s + 18/7·49-s + 1.51·63-s + 0.733·67-s + 0.481·69-s + 11/9·81-s − 2.41·83-s − 2.57·87-s + 1.27·89-s − 3.58·101-s + 1.77·103-s − 2.51·107-s − 2·121-s + 0.0887·127-s + 3.16·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 2.35·141-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 640000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.319675113\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.319675113\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 18 T + 162 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 22 T + 242 T^{2} + 22 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.71610194983247008539543311844, −10.00682755962414974756417550010, −9.316103785241621741446524711051, −9.249558785995916752160956436642, −8.621293987761356855561274600126, −8.498915720021783486744701714882, −7.75915321252701140688166020057, −7.74947840708286835695235705331, −7.26816707798868962597348483819, −6.82598839600891967469272805235, −5.92584847778616959918884557427, −5.51521269017675763037071240772, −5.14844395191820144991081675203, −4.47686394719811505501406495279, −4.09137344249998786444012231244, −3.71031466964137012738153082018, −2.64624472074562449511516543052, −2.46717075532667968130867247014, −1.67669514643215980846346607477, −1.10328903528200959302623318360,
1.10328903528200959302623318360, 1.67669514643215980846346607477, 2.46717075532667968130867247014, 2.64624472074562449511516543052, 3.71031466964137012738153082018, 4.09137344249998786444012231244, 4.47686394719811505501406495279, 5.14844395191820144991081675203, 5.51521269017675763037071240772, 5.92584847778616959918884557427, 6.82598839600891967469272805235, 7.26816707798868962597348483819, 7.74947840708286835695235705331, 7.75915321252701140688166020057, 8.498915720021783486744701714882, 8.621293987761356855561274600126, 9.249558785995916752160956436642, 9.316103785241621741446524711051, 10.00682755962414974756417550010, 10.71610194983247008539543311844
