| L(s) = 1 | − 16·2-s + 192·4-s − 155·5-s − 2.23e3·7-s − 2.04e3·8-s + 2.48e3·10-s + 3.29e3·11-s − 1.34e4·13-s + 3.58e4·14-s + 2.04e4·16-s + 3.22e4·17-s − 1.37e4·19-s − 2.97e4·20-s − 5.27e4·22-s + 8.25e4·23-s + 1.38e4·25-s + 2.14e5·26-s − 4.29e5·28-s + 1.27e4·29-s + 2.58e5·31-s − 1.96e5·32-s − 5.16e5·34-s + 3.46e5·35-s − 1.49e5·37-s + 2.19e5·38-s + 3.17e5·40-s − 3.39e5·41-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s − 0.554·5-s − 2.46·7-s − 1.41·8-s + 0.784·10-s + 0.746·11-s − 1.69·13-s + 3.48·14-s + 5/4·16-s + 1.59·17-s − 0.458·19-s − 0.831·20-s − 1.05·22-s + 1.41·23-s + 0.177·25-s + 2.39·26-s − 3.69·28-s + 0.0970·29-s + 1.56·31-s − 1.06·32-s − 2.25·34-s + 1.36·35-s − 0.484·37-s + 0.648·38-s + 0.784·40-s − 0.768·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 116964 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116964 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
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 | $C_1$ | \( ( 1 + p^{3} T )^{2} \) |
| 3 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 + p^{3} T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + 31 p T + 10178 T^{2} + 31 p^{8} T^{3} + p^{14} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 2238 T + 2775179 T^{2} + 2238 p^{7} T^{3} + p^{14} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 3295 T + 38080340 T^{2} - 3295 p^{7} T^{3} + p^{14} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 13427 T + 138664758 T^{2} + 13427 p^{7} T^{3} + p^{14} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 32256 T + 1073810905 T^{2} - 32256 p^{7} T^{3} + p^{14} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 82525 T + 5722043942 T^{2} - 82525 p^{7} T^{3} + p^{14} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 12749 T + 39176144 p^{2} T^{2} - 12749 p^{7} T^{3} + p^{14} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 258944 T + 64961539758 T^{2} - 258944 p^{7} T^{3} + p^{14} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 149260 T + 183707435838 T^{2} + 149260 p^{7} T^{3} + p^{14} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 339130 T - 4364095006 T^{2} + 339130 p^{7} T^{3} + p^{14} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 83869 T + 6409629042 p T^{2} + 83869 p^{7} T^{3} + p^{14} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 1471025 T + 1488141383726 T^{2} + 1471025 p^{7} T^{3} + p^{14} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 945643 T + 2428814565902 T^{2} - 945643 p^{7} T^{3} + p^{14} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 969009 T + 3139777837024 T^{2} - 969009 p^{7} T^{3} + p^{14} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 1506755 T + 5925806441724 T^{2} + 1506755 p^{7} T^{3} + p^{14} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 1848219 T + 11014380276458 T^{2} + 1848219 p^{7} T^{3} + p^{14} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 3417184 T + 7686276501674 T^{2} - 3417184 p^{7} T^{3} + p^{14} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 2499822 T + 17574764549507 T^{2} + 2499822 p^{7} T^{3} + p^{14} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 2636926 T + 33245149452510 T^{2} - 2636926 p^{7} T^{3} + p^{14} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 10059354 T + 77358130536910 T^{2} - 10059354 p^{7} T^{3} + p^{14} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 3506160 T + 657141629522 p T^{2} - 3506160 p^{7} T^{3} + p^{14} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 60758 p T + 73244272821042 T^{2} - 60758 p^{8} T^{3} + p^{14} T^{4} \) |
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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
−9.989346387171374681067834871672, −9.705440227361170725889707461215, −9.108811423603681556617396226436, −9.051815465380277817096659936920, −8.117109014439009281837302892955, −7.88209471229071628336750138834, −7.13886924206783903415881942568, −6.91197834775200878787473375995, −6.43524903782592662712392705405, −6.15493837029701718589545817382, −5.21152274844336313660180480606, −4.75249032495018224266641985611, −3.53278442229635270910413495585, −3.48886327048555533923477200394, −2.81577284765206954063061192499, −2.39603531107259905888402663096, −1.31316557630707060090343127160, −0.825925631549037790935745935516, 0, 0,
0.825925631549037790935745935516, 1.31316557630707060090343127160, 2.39603531107259905888402663096, 2.81577284765206954063061192499, 3.48886327048555533923477200394, 3.53278442229635270910413495585, 4.75249032495018224266641985611, 5.21152274844336313660180480606, 6.15493837029701718589545817382, 6.43524903782592662712392705405, 6.91197834775200878787473375995, 7.13886924206783903415881942568, 7.88209471229071628336750138834, 8.117109014439009281837302892955, 9.051815465380277817096659936920, 9.108811423603681556617396226436, 9.705440227361170725889707461215, 9.989346387171374681067834871672
