| L(s) = 1 | − 4·2-s + 12·4-s − 10·5-s + 7·7-s − 32·8-s + 40·10-s + 15·11-s + 49·13-s − 28·14-s + 80·16-s − 24·17-s + 64·19-s − 120·20-s − 60·22-s − 147·23-s + 75·25-s − 196·26-s + 84·28-s − 69·29-s − 143·31-s − 192·32-s + 96·34-s − 70·35-s + 358·37-s − 256·38-s + 320·40-s − 432·41-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s − 0.894·5-s + 0.377·7-s − 1.41·8-s + 1.26·10-s + 0.411·11-s + 1.04·13-s − 0.534·14-s + 5/4·16-s − 0.342·17-s + 0.772·19-s − 1.34·20-s − 0.581·22-s − 1.33·23-s + 3/5·25-s − 1.47·26-s + 0.566·28-s − 0.441·29-s − 0.828·31-s − 1.06·32-s + 0.484·34-s − 0.338·35-s + 1.59·37-s − 1.09·38-s + 1.26·40-s − 1.64·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 656100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 656100 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.546712758\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.546712758\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 - p T + 576 T^{2} - p^{4} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 15 T - 338 T^{2} - 15 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 49 T + 1938 T^{2} - 49 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 24 T + 8014 T^{2} + 24 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 64 T + 14253 T^{2} - 64 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 147 T + 14944 T^{2} + 147 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 69 T - 14702 T^{2} + 69 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 143 T + 61638 T^{2} + 143 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 358 T + 128946 T^{2} - 358 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 432 T + 144889 T^{2} + 432 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 628 T + 208710 T^{2} - 628 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 225 T + 117490 T^{2} - 225 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 231 T + 17572 T^{2} - 231 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 744 T + 548653 T^{2} + 744 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 802 T + 575154 T^{2} - 802 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 206 T + 502110 T^{2} + 206 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 1329 T + 971440 T^{2} - 1329 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 244 T + 722502 T^{2} - 244 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 1474 T + 1172766 T^{2} - 1474 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 858 T + 1268446 T^{2} + 858 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 2517 T + 2992660 T^{2} - 2517 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 1852 T + 2352258 T^{2} - 1852 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−9.969478590852703694641808933314, −9.519885366494521829896324445280, −9.204063776884823575025082631255, −8.857244552699823197570385872490, −8.195118144285691263039006174257, −8.144981495319917632185612424645, −7.51709222574424537408263345996, −7.44369821414621486409123408368, −6.65973624902526493822301539850, −6.32743771108083811905888615619, −5.88719871342586247186214346956, −5.30068512256230777133335653490, −4.62303100941955069519237306716, −3.98695107657464917795158898472, −3.58537224363169063145317138205, −3.08940643945435913313356990639, −2.08991145005058285663556025603, −1.84666122178262248429942192120, −0.77632017032405887554777541686, −0.62651600172808131262431162265,
0.62651600172808131262431162265, 0.77632017032405887554777541686, 1.84666122178262248429942192120, 2.08991145005058285663556025603, 3.08940643945435913313356990639, 3.58537224363169063145317138205, 3.98695107657464917795158898472, 4.62303100941955069519237306716, 5.30068512256230777133335653490, 5.88719871342586247186214346956, 6.32743771108083811905888615619, 6.65973624902526493822301539850, 7.44369821414621486409123408368, 7.51709222574424537408263345996, 8.144981495319917632185612424645, 8.195118144285691263039006174257, 8.857244552699823197570385872490, 9.204063776884823575025082631255, 9.519885366494521829896324445280, 9.969478590852703694641808933314
