| L(s) = 1 | + 6·3-s − 2·7-s + 27·9-s − 74·11-s − 98·13-s + 78·17-s + 80·19-s − 12·21-s + 40·23-s + 108·27-s + 50·29-s + 12·31-s − 444·33-s + 34·37-s − 588·39-s + 344·41-s + 216·43-s + 876·47-s + 478·49-s + 468·51-s + 634·53-s + 480·57-s − 666·59-s + 244·61-s − 54·63-s − 980·67-s + 240·69-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.107·7-s + 9-s − 2.02·11-s − 2.09·13-s + 1.11·17-s + 0.965·19-s − 0.124·21-s + 0.362·23-s + 0.769·27-s + 0.320·29-s + 0.0695·31-s − 2.34·33-s + 0.151·37-s − 2.41·39-s + 1.31·41-s + 0.766·43-s + 2.71·47-s + 1.39·49-s + 1.28·51-s + 1.64·53-s + 1.11·57-s − 1.46·59-s + 0.512·61-s − 0.107·63-s − 1.78·67-s + 0.418·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440000 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.704004542\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.704004542\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $D_{4}$ | \( 1 + 2 T - 474 T^{2} + 2 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 74 T + 3902 T^{2} + 74 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 98 T + 6666 T^{2} + 98 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 78 T + 8122 T^{2} - 78 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 80 T + 7062 T^{2} - 80 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 40 T + 16478 T^{2} - 40 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 50 T + 43082 T^{2} - 50 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 12 T + 17822 T^{2} - 12 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 34 T + 20970 T^{2} - 34 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 344 T + 162782 T^{2} - 344 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 108 T + p^{3} T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 - 876 T + 374206 T^{2} - 876 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 634 T + 398114 T^{2} - 634 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 666 T + 211918 T^{2} + 666 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 4 p T + 336750 T^{2} - 4 p^{4} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 980 T + 692502 T^{2} + 980 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 308 T + 553262 T^{2} + 308 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 1412 T + 1090194 T^{2} - 1412 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 1052 T + 1237470 T^{2} + 1052 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 248 T + 1125926 T^{2} - 248 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 684 T + 1229686 T^{2} - 684 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 1840 T + 2539650 T^{2} - 1840 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−9.372783420025994038417886062849, −9.313096713486356661811854610120, −8.803143652373925389246898097888, −8.340661919045081631091633034023, −7.68880235023874252580015271091, −7.48571793024379778994513480307, −7.42556424170616120194743333112, −7.14546779096170270080543260090, −6.02227217131908577254820088249, −5.82023630005094069514386690949, −5.22506420067414349624115897612, −4.90238083442971768066309319949, −4.44863420620650766132696366503, −3.84482372911806826403751293425, −3.13851992655724435334420760100, −2.84867373886782349622594371706, −2.31593724291088380612282652879, −2.19081336113120710667779492294, −0.902478242542447817355648283085, −0.58179064155895648041379189159,
0.58179064155895648041379189159, 0.902478242542447817355648283085, 2.19081336113120710667779492294, 2.31593724291088380612282652879, 2.84867373886782349622594371706, 3.13851992655724435334420760100, 3.84482372911806826403751293425, 4.44863420620650766132696366503, 4.90238083442971768066309319949, 5.22506420067414349624115897612, 5.82023630005094069514386690949, 6.02227217131908577254820088249, 7.14546779096170270080543260090, 7.42556424170616120194743333112, 7.48571793024379778994513480307, 7.68880235023874252580015271091, 8.340661919045081631091633034023, 8.803143652373925389246898097888, 9.313096713486356661811854610120, 9.372783420025994038417886062849
