| L(s) = 1 | − 28·7-s + 216·11-s + 1.37e4·23-s + 2.94e4·25-s − 3.49e4·29-s − 1.58e5·37-s + 2.15e5·43-s + 1.60e5·49-s + 1.58e5·53-s − 5.11e5·67-s − 1.64e6·71-s − 6.04e3·77-s − 2.82e6·79-s + 2.72e6·107-s − 1.03e5·109-s − 4.49e5·113-s − 3.89e6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s − 3.85e5·161-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | − 0.0816·7-s + 0.162·11-s + 1.13·23-s + 1.88·25-s − 1.43·29-s − 3.13·37-s + 2.70·43-s + 1.36·49-s + 1.06·53-s − 1.69·67-s − 4.60·71-s − 0.0132·77-s − 5.73·79-s + 2.22·107-s − 0.0798·109-s − 0.311·113-s − 2.19·121-s − 0.0922·161-s − 0.387·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(7-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+3)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.01541206783\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.01541206783\) |
| \(L(4)\) |
|
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 | | \( 1 \) |
| 7 | $D_{4}$ | \( 1 + 4 p T - 3258 p^{2} T^{2} + 4 p^{7} T^{3} + p^{12} T^{4} \) |
| good | 5 | $C_2^2 \wr C_2$ | \( 1 - 29476 T^{2} + 19858518 p^{2} T^{4} - 29476 p^{12} T^{6} + p^{24} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 - 108 T + 1965494 T^{2} - 108 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 13 | $C_2^2 \wr C_2$ | \( 1 + 1871900 T^{2} + 9531055477926 T^{4} + 1871900 p^{12} T^{6} + p^{24} T^{8} \) |
| 17 | $C_2^2 \wr C_2$ | \( 1 - 2295172 T^{2} + 1159564650619014 T^{4} - 2295172 p^{12} T^{6} + p^{24} T^{8} \) |
| 19 | $C_2^2 \wr C_2$ | \( 1 - 104314852 T^{2} + 6884640277857702 T^{4} - 104314852 p^{12} T^{6} + p^{24} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 - 6876 T + 282602918 T^{2} - 6876 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 29 | $D_{4}$ | \( ( 1 + 17460 T + 809082326 T^{2} + 17460 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 31 | $C_2^2 \wr C_2$ | \( 1 - 1008499972 T^{2} + 242873149828974342 T^{4} - 1008499972 p^{12} T^{6} + p^{24} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 + 79468 T + 6696018678 T^{2} + 79468 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 41 | $C_2^2 \wr C_2$ | \( 1 - 16302927556 T^{2} + \)\(10\!\cdots\!30\)\( T^{4} - 16302927556 p^{12} T^{6} + p^{24} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 107540 T + 7789273398 T^{2} - 107540 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 47 | $C_2^2 \wr C_2$ | \( 1 - 17280954244 T^{2} + \)\(23\!\cdots\!42\)\( T^{4} - 17280954244 p^{12} T^{6} + p^{24} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 - 79020 T + 33087355958 T^{2} - 79020 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 59 | $C_2^2 \wr C_2$ | \( 1 + 64559435228 T^{2} + \)\(32\!\cdots\!42\)\( T^{4} + 64559435228 p^{12} T^{6} + p^{24} T^{8} \) |
| 61 | $C_2^2 \wr C_2$ | \( 1 - 151438620772 T^{2} + \)\(10\!\cdots\!42\)\( T^{4} - 151438620772 p^{12} T^{6} + p^{24} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 + 255500 T + 196604609238 T^{2} + 255500 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 71 | $D_{4}$ | \( ( 1 + 824292 T + 371677978982 T^{2} + 824292 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 73 | $C_2^2 \wr C_2$ | \( 1 - 444319456708 T^{2} + \)\(93\!\cdots\!34\)\( T^{4} - 444319456708 p^{12} T^{6} + p^{24} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 1413884 T + 940532478150 T^{2} + 1413884 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 83 | $C_2^2 \wr C_2$ | \( 1 - 665529144292 T^{2} + \)\(30\!\cdots\!82\)\( T^{4} - 665529144292 p^{12} T^{6} + p^{24} T^{8} \) |
| 89 | $C_2^2 \wr C_2$ | \( 1 - 560004833092 T^{2} + \)\(18\!\cdots\!82\)\( T^{4} - 560004833092 p^{12} T^{6} + p^{24} T^{8} \) |
| 97 | $C_2^2 \wr C_2$ | \( 1 - 1385063966980 T^{2} + \)\(92\!\cdots\!26\)\( T^{4} - 1385063966980 p^{12} T^{6} + p^{24} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.58989695506760152645383977102, −7.25425807872047621303608459950, −7.17715441605047275965445842325, −7.10025392748025353649055288840, −6.73281885324654444053932339489, −6.34959389465520676177587165146, −5.81443659790569731357120238149, −5.75883812462203583254822456347, −5.74272940246660741642203798957, −5.22140863277172762888309809449, −4.90977360944895535842593453319, −4.72730631354679405205188644901, −4.10967036255262196835928049190, −4.06469443131667899925123240301, −4.00206069091803432701823480935, −3.12778326239087435934720874586, −3.03930089519379015658607891432, −2.92793814899366320016463955375, −2.53599180439434599846638417534, −1.95900194317880488287055830507, −1.62748656664250865351096782681, −1.31436286762166872621569283534, −1.08349506444679878847003780177, −0.56825249501369106134874756081, −0.01622757475400336676967216598,
0.01622757475400336676967216598, 0.56825249501369106134874756081, 1.08349506444679878847003780177, 1.31436286762166872621569283534, 1.62748656664250865351096782681, 1.95900194317880488287055830507, 2.53599180439434599846638417534, 2.92793814899366320016463955375, 3.03930089519379015658607891432, 3.12778326239087435934720874586, 4.00206069091803432701823480935, 4.06469443131667899925123240301, 4.10967036255262196835928049190, 4.72730631354679405205188644901, 4.90977360944895535842593453319, 5.22140863277172762888309809449, 5.74272940246660741642203798957, 5.75883812462203583254822456347, 5.81443659790569731357120238149, 6.34959389465520676177587165146, 6.73281885324654444053932339489, 7.10025392748025353649055288840, 7.17715441605047275965445842325, 7.25425807872047621303608459950, 7.58989695506760152645383977102
