| L(s) = 1 | + 8·7-s − 10·9-s + 132·17-s − 264·23-s − 25·25-s + 304·31-s + 876·41-s − 408·47-s − 638·49-s − 80·63-s − 864·71-s − 724·73-s − 320·79-s − 629·81-s − 1.62e3·89-s + 2.21e3·97-s − 2.07e3·113-s + 1.05e3·119-s + 2.51e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 1.32e3·153-s + 157-s + ⋯ |
| L(s) = 1 | + 0.431·7-s − 0.370·9-s + 1.88·17-s − 2.39·23-s − 1/5·25-s + 1.76·31-s + 3.33·41-s − 1.26·47-s − 1.86·49-s − 0.159·63-s − 1.44·71-s − 1.16·73-s − 0.455·79-s − 0.862·81-s − 1.92·89-s + 2.31·97-s − 1.72·113-s + 0.813·119-s + 1.89·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s − 0.697·153-s + 0.000508·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.433466473\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.433466473\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 10 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p^{3} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 2518 T^{2} + p^{6} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 1030 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 66 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 3718 T^{2} + p^{6} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 132 T + p^{3} T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 40678 T^{2} + p^{6} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 152 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 100150 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 438 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 157990 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 204 T + p^{3} T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 248470 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 234358 T^{2} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 359642 T^{2} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 447050 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 432 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 362 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 160 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 1138390 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 810 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 1106 T + p^{3} T^{2} )^{2} \) |
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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
−9.922209730572301062251873999987, −9.258253447272109001920048394046, −8.390003232217146344175375369858, −8.310445717514221102856939796935, −7.964986002609789878754630899586, −7.46739943291890699445640228728, −7.32260100981730608015028871905, −6.30180583583198474534969304251, −6.25028994028918074745659101015, −5.70296392808869601463585874376, −5.53260172383680491013297982608, −4.63203746638670448118391121580, −4.46099514712251998840508123575, −3.96525022484581501747524334019, −3.20304566533653100763421543424, −2.98284293069613874450983070645, −2.25801337125959235671303844648, −1.65595392011314236820540630010, −1.09443746676811413137401110765, −0.39504804746985530018064498754,
0.39504804746985530018064498754, 1.09443746676811413137401110765, 1.65595392011314236820540630010, 2.25801337125959235671303844648, 2.98284293069613874450983070645, 3.20304566533653100763421543424, 3.96525022484581501747524334019, 4.46099514712251998840508123575, 4.63203746638670448118391121580, 5.53260172383680491013297982608, 5.70296392808869601463585874376, 6.25028994028918074745659101015, 6.30180583583198474534969304251, 7.32260100981730608015028871905, 7.46739943291890699445640228728, 7.964986002609789878754630899586, 8.310445717514221102856939796935, 8.390003232217146344175375369858, 9.258253447272109001920048394046, 9.922209730572301062251873999987
