| L(s) = 1 | + 2-s + 6·3-s + 4-s + 10·5-s + 6·6-s − 14·7-s + 9·8-s + 27·9-s + 10·10-s − 22·11-s + 6·12-s − 22·13-s − 14·14-s + 60·15-s − 47·16-s + 116·17-s + 27·18-s + 102·19-s + 10·20-s − 84·21-s − 22·22-s + 260·23-s + 54·24-s + 75·25-s − 22·26-s + 108·27-s − 14·28-s + ⋯ |
| L(s) = 1 | + 0.353·2-s + 1.15·3-s + 1/8·4-s + 0.894·5-s + 0.408·6-s − 0.755·7-s + 0.397·8-s + 9-s + 0.316·10-s − 0.603·11-s + 0.144·12-s − 0.469·13-s − 0.267·14-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 − 0.872·21-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 − 0.0944·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11025 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.447550952\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.447550952\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 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
−13.40546592608703245871921898326, −13.30346175272847329227780832686, −12.73831602913954178485104586965, −12.18250555097709804072936119577, −11.53088546756880426510015652679, −10.67678270051248069312468546548, −10.24689496600514792997960725745, −9.750487568024786448080816428638, −9.153033322238177666395213432806, −8.959543556742975218076338319296, −7.920982822273216246053602816816, −7.35197583697436006371302737442, −7.02876736555687686837865597329, −6.13911297908769492799448079281, −5.17693363736470717283113350431, −4.97877979516232912120258346709, −3.60040878612527266076080746053, −3.08354544168682833867216834671, −2.36836063550997595589839875669, −1.18419157274904927371218082605,
1.18419157274904927371218082605, 2.36836063550997595589839875669, 3.08354544168682833867216834671, 3.60040878612527266076080746053, 4.97877979516232912120258346709, 5.17693363736470717283113350431, 6.13911297908769492799448079281, 7.02876736555687686837865597329, 7.35197583697436006371302737442, 7.920982822273216246053602816816, 8.959543556742975218076338319296, 9.153033322238177666395213432806, 9.750487568024786448080816428638, 10.24689496600514792997960725745, 10.67678270051248069312468546548, 11.53088546756880426510015652679, 12.18250555097709804072936119577, 12.73831602913954178485104586965, 13.30346175272847329227780832686, 13.40546592608703245871921898326
