| L(s) = 1 | − 32·4-s + 968·7-s − 6.73e3·13-s + 1.02e3·16-s + 1.14e4·19-s − 3.09e4·28-s − 7.95e4·31-s − 1.05e5·37-s − 7.60e3·43-s + 4.67e5·49-s + 2.15e5·52-s + 2.65e4·61-s − 3.27e4·64-s − 3.37e5·67-s − 4.72e5·73-s − 3.67e5·76-s − 7.02e4·79-s − 6.52e6·91-s + 6.42e5·97-s − 3.98e6·103-s + 3.88e5·109-s + 9.91e5·112-s + 1.74e6·121-s + 2.54e6·124-s + 127-s + 131-s + 1.11e7·133-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 2.82·7-s − 3.06·13-s + 1/4·16-s + 1.67·19-s − 1.41·28-s − 2.67·31-s − 2.07·37-s − 0.0955·43-s + 3.97·49-s + 1.53·52-s + 0.116·61-s − 1/8·64-s − 1.12·67-s − 1.21·73-s − 0.837·76-s − 0.142·79-s − 8.65·91-s + 0.704·97-s − 3.64·103-s + 0.300·109-s + 0.705·112-s + 0.985·121-s + 1.33·124-s + 4.72·133-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 202500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 202500 ^{s/2} \, \Gamma_{\C}(s+3)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.0001076726922\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0001076726922\) |
| \(L(4)\) |
|
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 | $C_2$ | \( 1 + p^{5} T^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2$ | \( ( 1 - 484 T + p^{6} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 1745714 T^{2} + p^{12} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 3368 T + p^{6} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 48274976 T^{2} + p^{12} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 5744 T + p^{6} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 284666690 T^{2} + p^{12} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 327940544 T^{2} + p^{12} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 39796 T + p^{6} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 52526 T + p^{6} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 8128061984 T^{2} + p^{12} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 3800 T + p^{6} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 15661450658 T^{2} + p^{12} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 12666935680 T^{2} + p^{12} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 21940729490 T^{2} + p^{12} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 13250 T + p^{6} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 168968 T + p^{6} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 26256724990 T^{2} + p^{12} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 236144 T + p^{6} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 35116 T + p^{6} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 653760187346 T^{2} + p^{12} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 977236742720 T^{2} + p^{12} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 321424 T + p^{6} T^{2} )^{2} \) |
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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
−10.49648953821918090087594765258, −9.700944834260700688310236305897, −9.570777892166045924374751290871, −8.821400692493894137392877716640, −8.651751680574406803009365971516, −7.79532032297496981636789602243, −7.69198312777834482587600532390, −7.21255387840803199088719846017, −7.10936171125943187962814236950, −5.82255638763566398631356230058, −5.12756626901347607236435244318, −5.10858422290881220503771753251, −4.97246406253046099714689393971, −4.19280308969180733019412290320, −3.60136241162053389919335700500, −2.70822628848661082135491629480, −2.19950392581183444509625105230, −1.53332702923560027312757993144, −1.31597678605706912600317125889, −0.00215436408347798118770047268,
0.00215436408347798118770047268, 1.31597678605706912600317125889, 1.53332702923560027312757993144, 2.19950392581183444509625105230, 2.70822628848661082135491629480, 3.60136241162053389919335700500, 4.19280308969180733019412290320, 4.97246406253046099714689393971, 5.10858422290881220503771753251, 5.12756626901347607236435244318, 5.82255638763566398631356230058, 7.10936171125943187962814236950, 7.21255387840803199088719846017, 7.69198312777834482587600532390, 7.79532032297496981636789602243, 8.651751680574406803009365971516, 8.821400692493894137392877716640, 9.570777892166045924374751290871, 9.700944834260700688310236305897, 10.49648953821918090087594765258
