| L(s) = 1 | + 2·3-s + 4·7-s + 9-s − 12·19-s + 8·21-s + 6·25-s + 2·27-s + 4·29-s − 12·31-s + 16·37-s + 20·47-s + 9·49-s + 4·53-s − 24·57-s + 8·59-s + 4·63-s + 12·75-s + 4·81-s + 12·83-s + 8·87-s − 24·93-s − 12·103-s − 8·109-s + 32·111-s + 14·113-s − 15·121-s + 127-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1.51·7-s + 1/3·9-s − 2.75·19-s + 1.74·21-s + 6/5·25-s + 0.384·27-s + 0.742·29-s − 2.15·31-s + 2.63·37-s + 2.91·47-s + 9/7·49-s + 0.549·53-s − 3.17·57-s + 1.04·59-s + 0.503·63-s + 1.38·75-s + 4/9·81-s + 1.31·83-s + 0.857·87-s − 2.48·93-s − 1.18·103-s − 0.766·109-s + 3.03·111-s + 1.31·113-s − 1.36·121-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 529984 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529984 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.425094978\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.425094978\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 3 | $C_2$$\times$$C_2$ | \( ( 1 - p T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 15 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 - 7 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 63 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 - 9 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 31 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 71 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 55 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 25 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 110 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 41 T^{2} + p^{2} T^{4} \) |
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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
−8.591237776065458957815717100656, −8.186396113827957594491553827864, −7.64328335999930451686515524910, −7.35293630424942421864765130650, −6.73032923751594741389024075974, −6.23262449838928905777935770488, −5.69731013514640098640396991738, −5.08723474259735470485098049520, −4.57757998992851307994405067926, −4.07277619574273699897386794950, −3.82927678109542374848938982943, −2.65023581762938548003097866108, −2.50346250579294179784168117969, −1.90392920847006658130137912002, −0.940108646971592877119260915250,
0.940108646971592877119260915250, 1.90392920847006658130137912002, 2.50346250579294179784168117969, 2.65023581762938548003097866108, 3.82927678109542374848938982943, 4.07277619574273699897386794950, 4.57757998992851307994405067926, 5.08723474259735470485098049520, 5.69731013514640098640396991738, 6.23262449838928905777935770488, 6.73032923751594741389024075974, 7.35293630424942421864765130650, 7.64328335999930451686515524910, 8.186396113827957594491553827864, 8.591237776065458957815717100656
