| L(s) = 1 | − 14·7-s − 28·13-s − 16·19-s − 31·25-s + 70·31-s − 88·37-s + 44·43-s + 49·49-s − 40·61-s − 28·67-s + 178·73-s + 220·79-s + 392·91-s + 22·97-s − 44·103-s + 104·109-s + 161·121-s + 127-s + 131-s + 224·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | − 2·7-s − 2.15·13-s − 0.842·19-s − 1.23·25-s + 2.25·31-s − 2.37·37-s + 1.02·43-s + 49-s − 0.655·61-s − 0.417·67-s + 2.43·73-s + 2.78·79-s + 4.30·91-s + 0.226·97-s − 0.427·103-s + 0.954·109-s + 1.33·121-s + 0.00787·127-s + 0.00763·131-s + 1.68·133-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2985984 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2985984 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.002034577566\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.002034577566\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 31 T^{2} + p^{4} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + p T + p^{2} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 161 T^{2} + p^{4} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 14 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 254 T^{2} + p^{4} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 8 T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 238 T^{2} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 1358 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 35 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 44 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 2066 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 22 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 1502 T^{2} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 5537 T^{2} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 6638 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 20 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 14 T + p^{2} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 5794 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 89 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 110 T + p^{2} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 13049 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 15518 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 11 T + p^{2} T^{2} )^{2} \) |
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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
−9.346697925421077517379847186802, −9.232916915134697024451354604470, −8.498941758067388952130340229759, −8.139984950584874711045919466631, −7.76522051770720956789519678151, −7.24876326889863685630432317786, −6.80090807467087225927817267625, −6.70096840900404468451968956647, −6.08393279566945503805256776516, −5.93040949419380816109570427535, −5.16953041399608696518823303478, −4.79385569092363362809055421669, −4.47627649820278110209360602000, −3.61757935907521937350659102984, −3.52313627937162632407918468023, −2.88081592126725158040821762489, −2.26713308065051085455337666034, −2.14477850723682371952637972800, −0.927196780929346804662308036628, −0.01391675100608439757174409572,
0.01391675100608439757174409572, 0.927196780929346804662308036628, 2.14477850723682371952637972800, 2.26713308065051085455337666034, 2.88081592126725158040821762489, 3.52313627937162632407918468023, 3.61757935907521937350659102984, 4.47627649820278110209360602000, 4.79385569092363362809055421669, 5.16953041399608696518823303478, 5.93040949419380816109570427535, 6.08393279566945503805256776516, 6.70096840900404468451968956647, 6.80090807467087225927817267625, 7.24876326889863685630432317786, 7.76522051770720956789519678151, 8.139984950584874711045919466631, 8.498941758067388952130340229759, 9.232916915134697024451354604470, 9.346697925421077517379847186802
