L(s) = 1 | + 4·4-s + 6·11-s + 12·16-s − 4·19-s + 6·29-s − 8·31-s + 24·41-s + 24·44-s − 49-s + 16·61-s + 32·64-s − 16·76-s + 2·79-s − 24·89-s − 12·101-s + 14·109-s + 24·116-s + 5·121-s − 32·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + ⋯ |
L(s) = 1 | + 2·4-s + 1.80·11-s + 3·16-s − 0.917·19-s + 1.11·29-s − 1.43·31-s + 3.74·41-s + 3.61·44-s − 1/7·49-s + 2.04·61-s + 4·64-s − 1.83·76-s + 0.225·79-s − 2.54·89-s − 1.19·101-s + 1.34·109-s + 2.22·116-s + 5/11·121-s − 2.87·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2480625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2480625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.336109469\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.336109469\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
good | 2 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 193 T^{2} + p^{2} T^{4} \) |
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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
−9.478365874869076816654096198730, −9.399349594446858084312927631376, −8.752835688899261381175208934640, −8.476032780663399742993589289873, −7.75858576815090926024565943630, −7.71096125778843593527190079586, −6.99040980534900117286456856642, −6.88226149251641680396258570888, −6.40995117947110185873739957064, −6.18613744789754897364081728834, −5.58416559008108437440761062076, −5.44082946768127261295685161203, −4.28902274945067524053534047938, −4.22280382744029398440466937073, −3.60758126763942390625852856145, −3.08984098876031870664575516273, −2.38979472144677827315004858676, −2.22062672683477142719393612124, −1.37921665283021032989857998998, −0.984811152605994644250568524101,
0.984811152605994644250568524101, 1.37921665283021032989857998998, 2.22062672683477142719393612124, 2.38979472144677827315004858676, 3.08984098876031870664575516273, 3.60758126763942390625852856145, 4.22280382744029398440466937073, 4.28902274945067524053534047938, 5.44082946768127261295685161203, 5.58416559008108437440761062076, 6.18613744789754897364081728834, 6.40995117947110185873739957064, 6.88226149251641680396258570888, 6.99040980534900117286456856642, 7.71096125778843593527190079586, 7.75858576815090926024565943630, 8.476032780663399742993589289873, 8.752835688899261381175208934640, 9.399349594446858084312927631376, 9.478365874869076816654096198730