| L(s) = 1 | + 9·5-s + 7-s + 63·11-s + 28·13-s − 144·17-s + 196·19-s + 126·23-s + 125·25-s − 126·29-s + 259·31-s + 9·35-s + 772·37-s − 450·41-s + 34·43-s − 54·47-s + 343·49-s + 1.38e3·53-s + 567·55-s + 180·59-s + 280·61-s + 252·65-s + 586·67-s − 1.00e3·71-s + 322·73-s + 63·77-s − 440·79-s + 999·83-s + ⋯ |
| L(s) = 1 | + 0.804·5-s + 0.0539·7-s + 1.72·11-s + 0.597·13-s − 2.05·17-s + 2.36·19-s + 1.14·23-s + 25-s − 0.806·29-s + 1.50·31-s + 0.0434·35-s + 3.43·37-s − 1.71·41-s + 0.120·43-s − 0.167·47-s + 49-s + 3.59·53-s + 1.39·55-s + 0.397·59-s + 0.587·61-s + 0.480·65-s + 1.06·67-s − 1.68·71-s + 0.516·73-s + 0.0932·77-s − 0.626·79-s + 1.32·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 104976 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 104976 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.610797507\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.610797507\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 9 T - 44 T^{2} - 9 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - T - 342 T^{2} - p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 63 T + 2638 T^{2} - 63 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 28 T - 1413 T^{2} - 28 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 72 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 98 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 126 T + 3709 T^{2} - 126 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 126 T - 8513 T^{2} + 126 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 259 T + 37290 T^{2} - 259 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 386 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 450 T + 133579 T^{2} + 450 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 34 T - 78351 T^{2} - 34 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 54 T - 100907 T^{2} + 54 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 693 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 180 T - 172979 T^{2} - 180 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 280 T - 148581 T^{2} - 280 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 586 T + 42633 T^{2} - 586 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 504 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 161 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 440 T - 299439 T^{2} + 440 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 999 T + 426214 T^{2} - 999 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 882 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 721 T - 392832 T^{2} - 721 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−11.27490921265456639126882499227, −11.25071340806086073110414500933, −10.44558533543398950030477442776, −9.914226892957083312216351734028, −9.414098107914900671332442148881, −9.241152309934714861075373218607, −8.624315827150434228927878409307, −8.394291053613700165521075090540, −7.25338042637745938199815086003, −7.18439776461159490621484937667, −6.42038369389618238693571707480, −6.24751173837841826179422627670, −5.43146876738338326187065693893, −5.01610469932105822388424082171, −4.15040617808194369854264522000, −3.88251553528622753647804949070, −2.82544755386384763771918236647, −2.39685750186238151734701212805, −1.14882764247013419221023359712, −1.01734501259156112813752983396,
1.01734501259156112813752983396, 1.14882764247013419221023359712, 2.39685750186238151734701212805, 2.82544755386384763771918236647, 3.88251553528622753647804949070, 4.15040617808194369854264522000, 5.01610469932105822388424082171, 5.43146876738338326187065693893, 6.24751173837841826179422627670, 6.42038369389618238693571707480, 7.18439776461159490621484937667, 7.25338042637745938199815086003, 8.394291053613700165521075090540, 8.624315827150434228927878409307, 9.241152309934714861075373218607, 9.414098107914900671332442148881, 9.914226892957083312216351734028, 10.44558533543398950030477442776, 11.25071340806086073110414500933, 11.27490921265456639126882499227
