| L(s) = 1 | + 2-s − 6·3-s + 4-s − 10·5-s − 6·6-s + 9·8-s + 27·9-s − 10·10-s − 22·11-s − 6·12-s + 22·13-s + 60·15-s − 47·16-s − 116·17-s + 27·18-s − 102·19-s − 10·20-s − 22·22-s + 260·23-s − 54·24-s + 75·25-s + 22·26-s − 108·27-s − 196·29-s + 60·30-s − 150·31-s − 103·32-s + ⋯ |
| L(s) = 1 | + 0.353·2-s − 1.15·3-s + 1/8·4-s − 0.894·5-s − 0.408·6-s + 0.397·8-s + 9-s − 0.316·10-s − 0.603·11-s − 0.144·12-s + 0.469·13-s + 1.03·15-s − 0.734·16-s − 1.65·17-s + 0.353·18-s − 1.23·19-s − 0.111·20-s − 0.213·22-s + 2.35·23-s − 0.459·24-s + 3/5·25-s + 0.165·26-s − 0.769·27-s − 1.25·29-s + 0.365·30-s − 0.869·31-s − 0.568·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 540225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 540225 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 5 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 7 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 - T - p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 2 p T + 2718 T^{2} + 2 p^{4} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 22 T + 4450 T^{2} - 22 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 116 T + 9030 T^{2} + 116 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 102 T + 13134 T^{2} + 102 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 260 T + 34734 T^{2} - 260 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 196 T + 20942 T^{2} + 196 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 150 T + 36542 T^{2} + 150 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 96 T + 82550 T^{2} + 96 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 176 T - 16914 T^{2} - 176 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 8 p T + 171958 T^{2} + 8 p^{4} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 560 T + 248606 T^{2} + 560 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 326 T + 204138 T^{2} - 326 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 844 T + 474182 T^{2} - 844 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 204 T + 455006 T^{2} - 204 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 104 T + 537670 T^{2} + 104 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 1670 T + 1384382 T^{2} - 1670 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 386 T + 152218 T^{2} - 386 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 888 T + 1007454 T^{2} + 888 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 928 T + 600710 T^{2} + 928 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 588 T + 1495334 T^{2} + 588 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 522 T + 291282 T^{2} + 522 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−9.788231791734121133244800726488, −9.411982426372493529729514189457, −8.745001715394745214491684725138, −8.583567694137355609637747489301, −8.065227875334943420692308503853, −7.34849276792725376954747804927, −6.95963816708906646539237970173, −6.75053180079583229876213778141, −6.43239991955889644454996613334, −5.52897637370693726493914237096, −5.19352521014216754559126703206, −4.89056365480134540174576408687, −4.18131232564101398669639312479, −4.02174794598969722967063116783, −3.30289378211385297672103041551, −2.45221579180846618649857924098, −1.91758650570189710748989195870, −1.05497619576561492344049950820, 0, 0,
1.05497619576561492344049950820, 1.91758650570189710748989195870, 2.45221579180846618649857924098, 3.30289378211385297672103041551, 4.02174794598969722967063116783, 4.18131232564101398669639312479, 4.89056365480134540174576408687, 5.19352521014216754559126703206, 5.52897637370693726493914237096, 6.43239991955889644454996613334, 6.75053180079583229876213778141, 6.95963816708906646539237970173, 7.34849276792725376954747804927, 8.065227875334943420692308503853, 8.583567694137355609637747489301, 8.745001715394745214491684725138, 9.411982426372493529729514189457, 9.788231791734121133244800726488
