| 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
| \(L(2)\) |
\(\approx\) |
\(2.708734088\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.708734088\) |
| \(L(\frac{5}{2})\) |
|
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
−11.35101465547225160168113612762, −11.14535436853415931281237418873, −10.35843789578670937672571032185, −10.10994286260179528909852532421, −9.536542371664218804918650587077, −9.317981175865497587191162646416, −8.200476841792196676564309207664, −7.982510118586333691003602260837, −7.63574343568576525627688197148, −7.59525428254638429934892376938, −6.37967877519881137023231582662, −6.07905970095647360429010729322, −5.35994749583048097088568535573, −5.00833592551767471313979028454, −4.31940513893412940706157992357, −3.60537207784656534884930927804, −2.87388971897964494457200464816, −2.68655719901948491506880040306, −1.06383263312562327872496585481, −0.77402054066306260397507537847,
0.77402054066306260397507537847, 1.06383263312562327872496585481, 2.68655719901948491506880040306, 2.87388971897964494457200464816, 3.60537207784656534884930927804, 4.31940513893412940706157992357, 5.00833592551767471313979028454, 5.35994749583048097088568535573, 6.07905970095647360429010729322, 6.37967877519881137023231582662, 7.59525428254638429934892376938, 7.63574343568576525627688197148, 7.982510118586333691003602260837, 8.200476841792196676564309207664, 9.317981175865497587191162646416, 9.536542371664218804918650587077, 10.10994286260179528909852532421, 10.35843789578670937672571032185, 11.14535436853415931281237418873, 11.35101465547225160168113612762
