| L(s) = 1 | + 20·7-s − 9·9-s + 28·17-s + 296·23-s + 234·25-s − 36·31-s + 756·41-s − 296·47-s − 386·49-s − 180·63-s + 1.08e3·71-s + 2.03e3·73-s − 772·79-s + 81·81-s + 764·89-s + 596·97-s − 1.63e3·103-s + 3.71e3·113-s + 560·119-s + 2.64e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 252·153-s + ⋯ |
| L(s) = 1 | + 1.07·7-s − 1/3·9-s + 0.399·17-s + 2.68·23-s + 1.87·25-s − 0.208·31-s + 2.87·41-s − 0.918·47-s − 1.12·49-s − 0.359·63-s + 1.80·71-s + 3.26·73-s − 1.09·79-s + 1/9·81-s + 0.909·89-s + 0.623·97-s − 1.56·103-s + 3.09·113-s + 0.431·119-s + 1.98·121-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 − 0.133·153-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 589824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 589824 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.907325621\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.907325621\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 22 T + p^{3} T^{2} )( 1 + 22 T + p^{3} T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 10 T + p^{3} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 2646 T^{2} + p^{6} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 22 p^{2} T^{2} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 14 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 13654 T^{2} + p^{6} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 148 T + p^{3} T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 43594 T^{2} + p^{6} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 18 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 32662 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 378 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 27610 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 148 T + p^{3} T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 168154 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 227574 T^{2} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 258598 T^{2} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 122662 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 540 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 1018 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 386 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 1131910 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 382 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 298 T + p^{3} T^{2} )^{2} \) |
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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
−10.13671447800914820061794110540, −9.638249388674601241210062040742, −9.194625054481607218614880262746, −8.924677586710793166844570445199, −8.402799002170471927296009377179, −8.082446623546456996175418170410, −7.50419738017026747256253793359, −7.24944305832535726074756285967, −6.52683299608085540246751455575, −6.41837798817966603507496796112, −5.42163118365804808692652217802, −5.29215424119792429895423143057, −4.68563721540095627185714986903, −4.50617432228308756946645252763, −3.50326325374502318435923655249, −3.13833079909511381037930898170, −2.54984579228122133414550522951, −1.87088262978212216674410874176, −0.928285175511649025560691921079, −0.819282415395064833124446378237,
0.819282415395064833124446378237, 0.928285175511649025560691921079, 1.87088262978212216674410874176, 2.54984579228122133414550522951, 3.13833079909511381037930898170, 3.50326325374502318435923655249, 4.50617432228308756946645252763, 4.68563721540095627185714986903, 5.29215424119792429895423143057, 5.42163118365804808692652217802, 6.41837798817966603507496796112, 6.52683299608085540246751455575, 7.24944305832535726074756285967, 7.50419738017026747256253793359, 8.082446623546456996175418170410, 8.402799002170471927296009377179, 8.924677586710793166844570445199, 9.194625054481607218614880262746, 9.638249388674601241210062040742, 10.13671447800914820061794110540
