| L(s) = 1 | − 52·11-s − 452·17-s − 268·19-s + 290·25-s − 1.98e3·41-s + 3.76e3·43-s + 962·49-s − 1.00e4·59-s − 1.60e4·67-s + 772·73-s − 4.46e3·83-s + 2.00e4·89-s + 1.74e4·97-s + 1.37e4·107-s + 2.42e3·113-s − 2.72e4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 5.61e4·169-s + 173-s + ⋯ |
| L(s) = 1 | − 0.429·11-s − 1.56·17-s − 0.742·19-s + 0.463·25-s − 1.18·41-s + 2.03·43-s + 0.400·49-s − 2.88·59-s − 3.56·67-s + 0.144·73-s − 0.648·83-s + 2.53·89-s + 1.85·97-s + 1.20·107-s + 0.190·113-s − 1.86·121-s + 6.20e−5·127-s + 5.82e−5·131-s + 5.32e−5·137-s + 5.17e−5·139-s + 4.50e−5·149-s + 4.38e−5·151-s + 4.05e−5·157-s + 3.76e−5·163-s + 3.58e−5·167-s + 1.96·169-s + 3.34e−5·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 82944 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 82944 ^{s/2} \, \Gamma_{\C}(s+2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.9082495790\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9082495790\) |
| \(L(3)\) |
|
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 | $C_2^2$ | \( 1 - 58 p T^{2} + p^{8} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 962 T^{2} + p^{8} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 26 T + p^{4} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 56162 T^{2} + p^{8} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 226 T + p^{4} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 134 T + p^{4} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 463682 T^{2} + p^{8} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 1298402 T^{2} + p^{8} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 311042 T^{2} + p^{8} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 629282 T^{2} + p^{8} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 994 T + p^{4} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 1882 T + p^{4} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 5320322 T^{2} + p^{8} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 1257122 T^{2} + p^{8} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 5018 T + p^{4} T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 23382242 T^{2} + p^{8} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 8006 T + p^{4} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 50512322 T^{2} + p^{8} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 386 T + p^{4} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 43766398 T^{2} + p^{8} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 2234 T + p^{4} T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 10046 T + p^{4} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 8738 T + p^{4} T^{2} )^{2} \) |
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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
−11.67665849275462766340248233243, −10.70873106546725583991521248168, −10.48658926396055988297128138684, −10.32531683688071186790515724878, −9.176731226241639431204382927539, −9.060108167658411913235310202978, −8.808027403618106079210782172761, −7.81311831188762893868191791016, −7.70860035248425707478130785121, −7.00722099114808520387883446828, −6.30120686533285719749997438109, −6.16730298654052484973307806064, −5.33002047474349156462590776306, −4.52081894242480347139486618509, −4.48121734903043566177245026951, −3.50517025112972445274058694529, −2.78292101673626730455490402483, −2.18447111856690174977388637853, −1.40482031235680000950625254296, −0.29577463829932304048185718482,
0.29577463829932304048185718482, 1.40482031235680000950625254296, 2.18447111856690174977388637853, 2.78292101673626730455490402483, 3.50517025112972445274058694529, 4.48121734903043566177245026951, 4.52081894242480347139486618509, 5.33002047474349156462590776306, 6.16730298654052484973307806064, 6.30120686533285719749997438109, 7.00722099114808520387883446828, 7.70860035248425707478130785121, 7.81311831188762893868191791016, 8.808027403618106079210782172761, 9.060108167658411913235310202978, 9.176731226241639431204382927539, 10.32531683688071186790515724878, 10.48658926396055988297128138684, 10.70873106546725583991521248168, 11.67665849275462766340248233243
