| L(s) = 1 | − 1.08e3·11-s − 1.67e3·19-s − 1.18e3·29-s + 8.51e3·31-s − 3.44e4·41-s + 2.58e4·49-s − 1.53e4·59-s − 6.94e4·61-s + 9.37e4·71-s + 1.53e5·79-s + 5.95e4·89-s − 2.25e4·101-s − 1.99e5·109-s + 5.52e5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 5.67e5·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | − 2.69·11-s − 1.06·19-s − 0.262·29-s + 1.59·31-s − 3.20·41-s + 1.53·49-s − 0.573·59-s − 2.39·61-s + 2.20·71-s + 2.77·79-s + 0.796·89-s − 0.220·101-s − 1.61·109-s + 3.43·121-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + 3.69e−6·149-s + 3.56e−6·151-s + 3.23e−6·157-s + 2.94e−6·163-s + 2.77e−6·167-s + 1.52·169-s + 2.54e−6·173-s + 2.33e−6·179-s + 2.26e−6·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 810000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810000 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.064722452\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.064722452\) |
| \(L(\frac{7}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 - 25870 T^{2} + p^{10} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 540 T + p^{5} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 567862 T^{2} + p^{10} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 2486878 T^{2} + p^{10} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 44 p T + p^{5} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 3970130 T^{2} + p^{10} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 594 T + p^{5} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4256 T + p^{5} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 138599110 T^{2} + p^{10} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 17226 T + p^{5} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 147606886 T^{2} + p^{10} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 457010398 T^{2} + p^{10} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 456374950 T^{2} + p^{10} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 7668 T + p^{5} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 34738 T + p^{5} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 2224486870 T^{2} + p^{10} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 46872 T + p^{5} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 418480658 T^{2} + p^{10} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 76912 T + p^{5} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 3292624630 T^{2} + p^{10} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 29754 T + p^{5} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 2193410110 T^{2} + p^{10} T^{4} \) |
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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
−9.599947340749524307770283161625, −9.196774681840523351866965026866, −8.574881558706222273816302232123, −8.259185652735669874548909017127, −7.86055489867944440276876455033, −7.74288631113524596028535444964, −6.96836180139172492263137897976, −6.60395728824762022453159269160, −6.18769132308770942256721161814, −5.55772141231612720390430138622, −5.09103315033385085129811913772, −4.94972971909911513305244724992, −4.34674078828028467722374994307, −3.69213032833222609434718102039, −3.05855191116269891629839567893, −2.73957655313444446460638721807, −2.12426163975535744952458126415, −1.75698941245322813925903634857, −0.73163819836755203867357653958, −0.26587996075051842240415559431,
0.26587996075051842240415559431, 0.73163819836755203867357653958, 1.75698941245322813925903634857, 2.12426163975535744952458126415, 2.73957655313444446460638721807, 3.05855191116269891629839567893, 3.69213032833222609434718102039, 4.34674078828028467722374994307, 4.94972971909911513305244724992, 5.09103315033385085129811913772, 5.55772141231612720390430138622, 6.18769132308770942256721161814, 6.60395728824762022453159269160, 6.96836180139172492263137897976, 7.74288631113524596028535444964, 7.86055489867944440276876455033, 8.259185652735669874548909017127, 8.574881558706222273816302232123, 9.196774681840523351866965026866, 9.599947340749524307770283161625
