| L(s) = 1 | + 8·2-s + 48·4-s − 100·7-s + 256·8-s + 540·11-s + 890·13-s − 800·14-s + 1.28e3·16-s − 492·17-s + 592·19-s + 4.32e3·22-s − 3.66e3·23-s + 7.12e3·26-s − 4.80e3·28-s + 5.70e3·29-s − 5.70e3·31-s + 6.14e3·32-s − 3.93e3·34-s + 1.13e4·37-s + 4.73e3·38-s + 1.54e4·41-s + 6.32e3·43-s + 2.59e4·44-s − 2.92e4·46-s + 7.80e3·47-s + 1.06e4·49-s + 4.27e4·52-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s − 0.771·7-s + 1.41·8-s + 1.34·11-s + 1.46·13-s − 1.09·14-s + 5/4·16-s − 0.412·17-s + 0.376·19-s + 1.90·22-s − 1.44·23-s + 2.06·26-s − 1.15·28-s + 1.25·29-s − 1.06·31-s + 1.06·32-s − 0.583·34-s + 1.35·37-s + 0.532·38-s + 1.43·41-s + 0.521·43-s + 2.01·44-s − 2.04·46-s + 0.515·47-s + 0.631·49-s + 2.19·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 202500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 202500 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(12.60077989\) |
| \(L(\frac12)\) |
\(\approx\) |
\(12.60077989\) |
| \(L(\frac{7}{2})\) |
|
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 | $C_1$ | \( ( 1 - p^{2} T )^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $D_{4}$ | \( 1 + 100 T - 615 T^{2} + 100 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 540 T + 248086 T^{2} - 540 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 890 T + 793695 T^{2} - 890 p^{5} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 492 T - 772670 T^{2} + 492 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 592 T + 4121589 T^{2} - 592 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 3660 T + 12548686 T^{2} + 3660 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 5700 T + 48997882 T^{2} - 5700 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 5708 T + 64485393 T^{2} + 5708 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 11300 T + 149454510 T^{2} - 11300 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 15420 T + 258100402 T^{2} - 15420 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 6320 T + 242261037 T^{2} - 6320 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 7800 T + 106610014 T^{2} - 7800 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 27828 T + 850018282 T^{2} - 27828 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 50520 T + 1968600982 T^{2} - 50520 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 29126 T + 1603768671 T^{2} + 29126 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 97400 T + 5071609653 T^{2} - 97400 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 6180 T + 3034897198 T^{2} - 6180 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 32900 T + 3887848086 T^{2} - 32900 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 7912 T + 1409684334 T^{2} - 7912 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 163464 T + 14190911110 T^{2} - 163464 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 164640 T + 13798144114 T^{2} - 164640 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 52430 T + 12995461155 T^{2} - 52430 p^{5} T^{3} + p^{10} T^{4} \) |
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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
−10.54057356582370531588045987280, −10.32627065676839019790681004342, −9.470577925391273374872187903890, −9.343495070138615490372798401507, −8.711957864695424859084550506951, −8.158645163664201900097422471645, −7.64929407619426633357096459919, −7.06221460948643103172531441905, −6.40747546547904609800258678830, −6.38112851004476586466620221545, −5.87211889786077358600574457941, −5.38459088033591376513774978725, −4.60521279005432961062346624182, −4.03908063726808314806988562901, −3.71001704702754381960406340571, −3.45770296519056312639079665941, −2.35416799302220547344757399583, −2.20553109247609398916413525286, −1.01314772470290716430014918384, −0.801178840532270072047280285403,
0.801178840532270072047280285403, 1.01314772470290716430014918384, 2.20553109247609398916413525286, 2.35416799302220547344757399583, 3.45770296519056312639079665941, 3.71001704702754381960406340571, 4.03908063726808314806988562901, 4.60521279005432961062346624182, 5.38459088033591376513774978725, 5.87211889786077358600574457941, 6.38112851004476586466620221545, 6.40747546547904609800258678830, 7.06221460948643103172531441905, 7.64929407619426633357096459919, 8.158645163664201900097422471645, 8.711957864695424859084550506951, 9.343495070138615490372798401507, 9.470577925391273374872187903890, 10.32627065676839019790681004342, 10.54057356582370531588045987280
