| L(s) = 1 | + 2·3-s + 3·9-s + 5·13-s + 3·17-s + 6·19-s + 6·23-s + 7·25-s + 10·27-s + 3·29-s − 15·37-s + 10·39-s − 9·41-s + 8·43-s − 7·49-s + 6·51-s + 6·53-s + 12·57-s − 12·59-s + 61-s − 6·67-s + 12·69-s + 6·71-s + 14·75-s + 8·79-s + 20·81-s + 6·87-s − 12·89-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 9-s + 1.38·13-s + 0.727·17-s + 1.37·19-s + 1.25·23-s + 7/5·25-s + 1.92·27-s + 0.557·29-s − 2.46·37-s + 1.60·39-s − 1.40·41-s + 1.21·43-s − 49-s + 0.840·51-s + 0.824·53-s + 1.58·57-s − 1.56·59-s + 0.128·61-s − 0.733·67-s + 1.44·69-s + 0.712·71-s + 1.61·75-s + 0.900·79-s + 20/9·81-s + 0.643·87-s − 1.27·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 692224 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 692224 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.083378886\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.083378886\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 13 | $C_2$ | \( 1 - 5 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T + T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 7 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 3 T - 20 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 15 T + 112 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 9 T + 68 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 12 T + 107 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 6 T + 83 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 12 T + 137 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 12 T + 145 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
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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
−10.35156224761615169015313377590, −10.05947621800990910940676282125, −9.393142522121543241355094837957, −9.081748904585127415739442859446, −8.719798006643904438057059673349, −8.451177385438783730373633503685, −7.981178807168000502414143638320, −7.48945524821787918253428785385, −6.87255681828692222734207562669, −6.84870111365291030756401628935, −6.16346685631927636428370955012, −5.48585094412207632847449169395, −4.92680899819577653089241091085, −4.77595904803463638852110189098, −3.73821556874204723903753598097, −3.44600387721085178412118898592, −3.06558804326536764668254183560, −2.52519132039434657488003662700, −1.34757788576870210394749588045, −1.19342195581952098885891454876,
1.19342195581952098885891454876, 1.34757788576870210394749588045, 2.52519132039434657488003662700, 3.06558804326536764668254183560, 3.44600387721085178412118898592, 3.73821556874204723903753598097, 4.77595904803463638852110189098, 4.92680899819577653089241091085, 5.48585094412207632847449169395, 6.16346685631927636428370955012, 6.84870111365291030756401628935, 6.87255681828692222734207562669, 7.48945524821787918253428785385, 7.981178807168000502414143638320, 8.451177385438783730373633503685, 8.719798006643904438057059673349, 9.081748904585127415739442859446, 9.393142522121543241355094837957, 10.05947621800990910940676282125, 10.35156224761615169015313377590
