L(s) = 1 | − 4-s + 5-s + 9-s + 16-s − 2·19-s − 20-s + 25-s − 2·29-s − 36-s − 2·41-s + 45-s + 49-s + 59-s − 64-s − 2·71-s + 2·76-s + 2·79-s + 80-s + 81-s − 2·95-s − 100-s + 2·116-s + ⋯ |
L(s) = 1 | − 4-s + 5-s + 9-s + 16-s − 2·19-s − 20-s + 25-s − 2·29-s − 36-s − 2·41-s + 45-s + 49-s + 59-s − 64-s − 2·71-s + 2·76-s + 2·79-s + 80-s + 81-s − 2·95-s − 100-s + 2·116-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 295 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 295 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7544110325\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7544110325\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 - T \) |
| 59 | \( 1 - T \) |
good | 2 | \( 1 + T^{2} \) |
| 3 | \( ( 1 - T )( 1 + T ) \) |
| 7 | \( ( 1 - T )( 1 + T ) \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( ( 1 + T )^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( ( 1 + T )^{2} \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( ( 1 + T )^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( ( 1 + T )^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( ( 1 - T )^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 97 | \( 1 + T^{2} \) |
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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
−12.30316135223849536748441314461, −10.73873661196391459901143738163, −10.06339346522991188767713250654, −9.233268245566741331454016685878, −8.423602640577405702990143688298, −7.08287576499635293686916466191, −5.95685201287422724050475808782, −4.87654148899374470308483011361, −3.83650161991916834317285358945, −1.88469320597866519526099075548,
1.88469320597866519526099075548, 3.83650161991916834317285358945, 4.87654148899374470308483011361, 5.95685201287422724050475808782, 7.08287576499635293686916466191, 8.423602640577405702990143688298, 9.233268245566741331454016685878, 10.06339346522991188767713250654, 10.73873661196391459901143738163, 12.30316135223849536748441314461