| L(s) = 1 | − 306·9-s − 308·13-s − 356·17-s + 8.22e3·29-s − 1.48e4·37-s + 1.45e4·41-s − 1.43e4·49-s − 6.44e4·53-s + 5.35e4·61-s + 3.70e4·73-s + 3.45e4·81-s − 2.15e5·89-s + 2.17e5·97-s + 1.18e5·101-s − 2.79e5·109-s + 8.60e4·113-s + 9.42e4·117-s − 2.54e5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 1.08e5·153-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | − 1.25·9-s − 0.505·13-s − 0.298·17-s + 1.81·29-s − 1.78·37-s + 1.35·41-s − 0.856·49-s − 3.15·53-s + 1.84·61-s + 0.814·73-s + 0.585·81-s − 2.87·89-s + 2.34·97-s + 1.15·101-s − 2.25·109-s + 0.634·113-s + 0.636·117-s − 1.58·121-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 + 0.376·153-s + 3.23e−6·157-s + 2.94e−6·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 640000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640000 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + 34 p^{2} T^{2} + p^{10} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 14394 T^{2} + p^{10} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 254822 T^{2} + p^{10} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 154 T + p^{5} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 178 T + p^{5} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 4019078 T^{2} + p^{10} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 5934266 T^{2} + p^{10} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 4110 T + p^{5} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 47289582 T^{2} + p^{10} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 7442 T + p^{5} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 7270 T + p^{5} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 26783614 T^{2} + p^{10} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 403777034 T^{2} + p^{10} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 32226 T + p^{5} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 270997718 T^{2} + p^{10} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 26770 T + p^{5} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 219594834 T^{2} + p^{10} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 681226622 T^{2} + p^{10} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 18534 T + p^{5} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 1369983522 T^{2} + p^{10} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 1693436786 T^{2} + p^{10} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 107590 T + p^{5} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 108838 T + p^{5} T^{2} )^{2} \) |
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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
−9.252851356153774228240289255362, −8.866798605919223634882271570379, −8.284035393377373457706167055292, −8.277326852667529856183105652561, −7.65069221314233039694038566858, −7.11410891863381568787892144022, −6.60010063590486351797700542276, −6.33419648230428698367492676986, −5.75823467708972874562632621680, −5.32730006076728036180106055448, −4.66567137537736186093221353866, −4.64608362656786922496775567525, −3.57876643013929531538553017778, −3.37919285303403460679555527296, −2.52961870248768757062550800202, −2.47520048147197230565708563054, −1.53172132068198933094610362192, −0.969060362544447441382930510996, 0, 0,
0.969060362544447441382930510996, 1.53172132068198933094610362192, 2.47520048147197230565708563054, 2.52961870248768757062550800202, 3.37919285303403460679555527296, 3.57876643013929531538553017778, 4.64608362656786922496775567525, 4.66567137537736186093221353866, 5.32730006076728036180106055448, 5.75823467708972874562632621680, 6.33419648230428698367492676986, 6.60010063590486351797700542276, 7.11410891863381568787892144022, 7.65069221314233039694038566858, 8.277326852667529856183105652561, 8.284035393377373457706167055292, 8.866798605919223634882271570379, 9.252851356153774228240289255362
