| L(s) = 1 | + 12·3-s − 52·7-s + 138·9-s + 560·11-s − 1.38e3·13-s − 148·17-s − 1.00e3·19-s − 624·21-s + 2.45e3·23-s + 4.50e3·27-s + 1.34e3·29-s − 2.24e3·31-s + 6.72e3·33-s + 5.94e3·37-s − 1.66e4·39-s + 2.30e4·41-s − 1.76e4·43-s + 2.90e3·47-s − 2.69e4·49-s − 1.77e3·51-s + 5.41e3·53-s − 1.20e4·57-s + 6.25e4·59-s + 1.41e4·61-s − 7.17e3·63-s + 8.54e4·67-s + 2.94e4·69-s + ⋯ |
| L(s) = 1 | + 0.769·3-s − 0.401·7-s + 0.567·9-s + 1.39·11-s − 2.27·13-s − 0.124·17-s − 0.635·19-s − 0.308·21-s + 0.966·23-s + 1.18·27-s + 0.295·29-s − 0.420·31-s + 1.07·33-s + 0.713·37-s − 1.75·39-s + 2.14·41-s − 1.45·43-s + 0.192·47-s − 1.60·49-s − 0.0956·51-s + 0.264·53-s − 0.489·57-s + 2.34·59-s + 0.485·61-s − 0.227·63-s + 2.32·67-s + 0.744·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40000 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.083047960\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.083047960\) |
| \(L(\frac{7}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $D_{4}$ | \( 1 - 4 p T + 2 p T^{2} - 4 p^{6} T^{3} + p^{10} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 52 T + 29646 T^{2} + 52 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 560 T + 381926 T^{2} - 560 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 1388 T + 1149918 T^{2} + 1388 p^{5} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 148 T - 795706 T^{2} + 148 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 1000 T + 4533462 T^{2} + 1000 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 2452 T + 6569198 T^{2} - 2452 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 1340 T - 8758306 T^{2} - 1340 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 2248 T + 57017022 T^{2} + 2248 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 5940 T + 123434318 T^{2} - 5940 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 23076 T + 352280470 T^{2} - 23076 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 17684 T + 312898614 T^{2} + 17684 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 2908 T + 56660030 T^{2} - 2908 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 5412 T + 693247822 T^{2} - 5412 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 62584 T + 2277965606 T^{2} - 62584 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 14108 T + 1110042462 T^{2} - 14108 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 85412 T + 4371910566 T^{2} - 85412 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 47208 T + 4011779662 T^{2} - 47208 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 924 p T + 4400780438 T^{2} - 924 p^{6} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 65904 T + 3994274078 T^{2} + 65904 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 108724 T + 10572459494 T^{2} + 108724 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 55020 T + 10818978262 T^{2} + 55020 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 147668 T + 11612429670 T^{2} + 147668 p^{5} T^{3} + p^{10} T^{4} \) |
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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
−11.86800061743356275809531161855, −11.30134249018345087786333026922, −10.97387429433173201191536389016, −10.00415779018511087319087965194, −9.672980452802737630158226458969, −9.630698738994095497353451024463, −8.773620991588071367408542170075, −8.467702404460738700238124906097, −7.77846085531316808252923452895, −7.12528680418014863275995431880, −6.79735548974311278354844971391, −6.39852578761321122935162318851, −5.32621770889830913245954720176, −4.87161598524438092448056669666, −4.14782071824382780316643413271, −3.66943912053539313552940196748, −2.61803087250231913778005489820, −2.46461740531942314071396190316, −1.37632882417086513983056556340, −0.52621287453404854682625772264,
0.52621287453404854682625772264, 1.37632882417086513983056556340, 2.46461740531942314071396190316, 2.61803087250231913778005489820, 3.66943912053539313552940196748, 4.14782071824382780316643413271, 4.87161598524438092448056669666, 5.32621770889830913245954720176, 6.39852578761321122935162318851, 6.79735548974311278354844971391, 7.12528680418014863275995431880, 7.77846085531316808252923452895, 8.467702404460738700238124906097, 8.773620991588071367408542170075, 9.630698738994095497353451024463, 9.672980452802737630158226458969, 10.00415779018511087319087965194, 10.97387429433173201191536389016, 11.30134249018345087786333026922, 11.86800061743356275809531161855
