| L(s) = 1 | − 4-s − 4·11-s + 16-s − 8·19-s + 2·29-s + 6·31-s + 6·41-s + 4·44-s − 49-s − 6·59-s + 10·61-s − 64-s − 8·71-s + 8·76-s + 4·79-s + 20·89-s + 8·109-s − 2·116-s − 10·121-s − 6·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 1.20·11-s + 1/4·16-s − 1.83·19-s + 0.371·29-s + 1.07·31-s + 0.937·41-s + 0.603·44-s − 1/7·49-s − 0.781·59-s + 1.28·61-s − 1/8·64-s − 0.949·71-s + 0.917·76-s + 0.450·79-s + 2.11·89-s + 0.766·109-s − 0.185·116-s − 0.909·121-s − 0.538·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9922500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9922500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.206515804\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.206515804\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| good | 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 33 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 3 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 157 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
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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
−8.902186558754098285799163241864, −8.441209391517775762363458788091, −8.056352970138622546857262447687, −7.917622718233711942763675610896, −7.52875966659886035518224209237, −6.93824515043766526694219211083, −6.51388161801017148728456699899, −6.36490624378062815697104960504, −5.62609604427547737567948586981, −5.62442206046540222346720984564, −4.85295102670311579135486305208, −4.71876505107304421530976423227, −4.16767890302369733025766307495, −3.96838017352575114936592203683, −3.08103096442649605749197867140, −2.98695522803573316007910573951, −2.17203488202553299466463835934, −2.04139695684873212623534625709, −1.03404631717384617889362410988, −0.38891980352997469944515553764,
0.38891980352997469944515553764, 1.03404631717384617889362410988, 2.04139695684873212623534625709, 2.17203488202553299466463835934, 2.98695522803573316007910573951, 3.08103096442649605749197867140, 3.96838017352575114936592203683, 4.16767890302369733025766307495, 4.71876505107304421530976423227, 4.85295102670311579135486305208, 5.62442206046540222346720984564, 5.62609604427547737567948586981, 6.36490624378062815697104960504, 6.51388161801017148728456699899, 6.93824515043766526694219211083, 7.52875966659886035518224209237, 7.917622718233711942763675610896, 8.056352970138622546857262447687, 8.441209391517775762363458788091, 8.902186558754098285799163241864
