| L(s) = 1 | − 4·4-s + 96·11-s + 16·16-s − 238·19-s − 60·29-s − 266·31-s − 312·41-s − 384·44-s − 470·49-s − 1.30e3·59-s + 922·61-s − 64·64-s − 1.80e3·71-s + 952·76-s + 2.75e3·79-s − 2.23e3·89-s − 696·101-s + 230·109-s + 240·116-s + 4.25e3·121-s + 1.06e3·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 2.63·11-s + 1/4·16-s − 2.87·19-s − 0.384·29-s − 1.54·31-s − 1.18·41-s − 1.31·44-s − 1.37·49-s − 2.88·59-s + 1.93·61-s − 1/8·64-s − 3.00·71-s + 1.43·76-s + 3.91·79-s − 2.65·89-s − 0.685·101-s + 0.202·109-s + 0.192·116-s + 3.19·121-s + 0.770·124-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1822500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1822500 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.3866883823\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3866883823\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 + 470 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 48 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 506 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 9097 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 119 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 21733 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 30 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 133 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 53782 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 156 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 151270 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 58610 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 110567 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 654 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 461 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 568402 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 900 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 282418 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 1375 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 306349 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 1116 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 1825090 T^{2} + p^{6} T^{4} \) |
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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
−9.485454875225765112714390274567, −8.887675221801157431580664837797, −8.771738635132350757688645017696, −8.473795059267402870868122414589, −7.81374164997460838477116146497, −7.47216640964386145702757784265, −6.72335399605076081555794648580, −6.46923232856222723031263078628, −6.44171571820989759679921351708, −5.77585821256637075775902146289, −5.22577570039353875229529536511, −4.64733371026879941904706558185, −4.16328379420592955352982963480, −4.00484449350700469693381153558, −3.52808310964517029907396081625, −2.92994813038829814199614497534, −1.90663120141482882917363461162, −1.78677493634525629688036978765, −1.15703228831611031278487762868, −0.14791326237995688907206672943,
0.14791326237995688907206672943, 1.15703228831611031278487762868, 1.78677493634525629688036978765, 1.90663120141482882917363461162, 2.92994813038829814199614497534, 3.52808310964517029907396081625, 4.00484449350700469693381153558, 4.16328379420592955352982963480, 4.64733371026879941904706558185, 5.22577570039353875229529536511, 5.77585821256637075775902146289, 6.44171571820989759679921351708, 6.46923232856222723031263078628, 6.72335399605076081555794648580, 7.47216640964386145702757784265, 7.81374164997460838477116146497, 8.473795059267402870868122414589, 8.771738635132350757688645017696, 8.887675221801157431580664837797, 9.485454875225765112714390274567
