| L(s) = 1 | + 4·4-s − 52·13-s + 12·16-s + 44·25-s + 272·43-s − 92·49-s − 208·52-s + 160·61-s + 32·64-s − 272·79-s + 176·100-s + 32·103-s − 448·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.69e3·169-s + 1.08e3·172-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | + 4-s − 4·13-s + 3/4·16-s + 1.75·25-s + 6.32·43-s − 1.87·49-s − 4·52-s + 2.62·61-s + 1/2·64-s − 3.44·79-s + 1.75·100-s + 0.310·103-s − 3.70·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 10·169-s + 6.32·172-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.594198429\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.594198429\) |
| \(L(2)\) |
|
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 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 + p T )^{4} \) |
| good | 5 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{4} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 + 46 T^{2} + p^{4} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 224 T^{2} + p^{4} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 560 T^{2} + p^{4} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 398 T^{2} + p^{4} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 770 T^{2} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 224 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 158 T^{2} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 862 T^{2} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 3290 T^{2} + p^{4} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 68 T + p^{2} T^{2} )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 + 2960 T^{2} + p^{4} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 880 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 6512 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 40 T + p^{2} T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 5378 T^{2} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 7040 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 10514 T^{2} + p^{4} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 68 T + p^{2} T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + 656 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 15194 T^{2} + p^{4} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 7154 T^{2} + p^{4} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.586928176925743399948997629972, −8.274190751033050832806578557328, −8.120352197279141904990986056728, −7.71011359015305937837173702935, −7.41963749202955463082599794438, −7.35577779826890514629786463907, −6.98232178259551108034420165955, −6.90575004551449815171182614079, −6.87081479479021349737109346310, −5.93786657546287248973852342141, −5.91180037368603887883469121921, −5.85908923151408457898085261261, −5.25441050470376310929718352724, −4.93105808837835451043479061466, −4.82981544770466272976278695237, −4.37583996444853714369888415489, −4.21073027387699933563512138064, −3.70769850109466793336107580651, −3.08103181168189744248362218624, −2.70172078550987140732687866695, −2.48469244808739337617783943361, −2.47904346888002131815549570179, −1.83181400181284102196711986356, −1.03252961054679696068066286711, −0.46465115923024752375314531020,
0.46465115923024752375314531020, 1.03252961054679696068066286711, 1.83181400181284102196711986356, 2.47904346888002131815549570179, 2.48469244808739337617783943361, 2.70172078550987140732687866695, 3.08103181168189744248362218624, 3.70769850109466793336107580651, 4.21073027387699933563512138064, 4.37583996444853714369888415489, 4.82981544770466272976278695237, 4.93105808837835451043479061466, 5.25441050470376310929718352724, 5.85908923151408457898085261261, 5.91180037368603887883469121921, 5.93786657546287248973852342141, 6.87081479479021349737109346310, 6.90575004551449815171182614079, 6.98232178259551108034420165955, 7.35577779826890514629786463907, 7.41963749202955463082599794438, 7.71011359015305937837173702935, 8.120352197279141904990986056728, 8.274190751033050832806578557328, 8.586928176925743399948997629972
