| L(s) = 1 | − 2·3-s + 34·5-s − 14·7-s + 24·9-s + 4·11-s + 74·13-s − 68·15-s − 140·17-s + 246·19-s + 28·21-s + 288·23-s + 644·25-s − 142·27-s + 68·29-s − 280·31-s − 8·33-s − 476·35-s + 164·37-s − 148·39-s + 244·41-s − 188·43-s + 816·45-s − 392·47-s + 147·49-s + 280·51-s + 504·53-s + 136·55-s + ⋯ |
| L(s) = 1 | − 0.384·3-s + 3.04·5-s − 0.755·7-s + 8/9·9-s + 0.109·11-s + 1.57·13-s − 1.17·15-s − 1.99·17-s + 2.97·19-s + 0.290·21-s + 2.61·23-s + 5.15·25-s − 1.01·27-s + 0.435·29-s − 1.62·31-s − 0.0422·33-s − 2.29·35-s + 0.728·37-s − 0.607·39-s + 0.929·41-s − 0.666·43-s + 2.70·45-s − 1.21·47-s + 3/7·49-s + 0.768·51-s + 1.30·53-s + 0.333·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3211264 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3211264 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(8.872097869\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.872097869\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + 2 T - 20 T^{2} + 2 p^{3} T^{3} + p^{6} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 34 T + 512 T^{2} - 34 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 4 T + 1214 T^{2} - 4 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 74 T + 5256 T^{2} - 74 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 140 T + 12998 T^{2} + 140 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 246 T + 28820 T^{2} - 246 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 288 T + 1934 p T^{2} - 288 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 68 T + 46466 T^{2} - 68 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 280 T + 77454 T^{2} + 280 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 164 T - 270 T^{2} - 164 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 244 T + 120278 T^{2} - 244 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 188 T + 143550 T^{2} + 188 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 392 T + 157310 T^{2} + 392 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 504 T + 259690 T^{2} - 504 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 190 T + 281108 T^{2} - 190 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 398 T + 480096 T^{2} + 398 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 368 T + 296694 T^{2} - 368 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 248 T + 717326 T^{2} - 248 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 204 T + 691238 T^{2} + 204 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 440 T + 792510 T^{2} + 440 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 1158 T + 1292812 T^{2} + 1158 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 1156 T + 1720790 T^{2} - 1156 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 428 T + 1022070 T^{2} + 428 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−9.367518156617666493058351888766, −9.072160984887336108171307850338, −8.594552209368300032470014171688, −7.948054728561044771508459080490, −7.13395138051234912277188883841, −6.99157516486026224546259109212, −6.70631457956841859847741614347, −6.34962742260225359731017650802, −5.77076904962778099198640475491, −5.55087138225434556210069399821, −5.32675523747015461635338011543, −4.83798570174114508009405095292, −4.20878185424881053505453524163, −3.59272122474116931512937375147, −2.88124283625549564823297638143, −2.82160621236201159023306935012, −1.98600659681252826107584278249, −1.46653673386621330952481902788, −1.24403967326633708499965898976, −0.67016935851971252498531363509,
0.67016935851971252498531363509, 1.24403967326633708499965898976, 1.46653673386621330952481902788, 1.98600659681252826107584278249, 2.82160621236201159023306935012, 2.88124283625549564823297638143, 3.59272122474116931512937375147, 4.20878185424881053505453524163, 4.83798570174114508009405095292, 5.32675523747015461635338011543, 5.55087138225434556210069399821, 5.77076904962778099198640475491, 6.34962742260225359731017650802, 6.70631457956841859847741614347, 6.99157516486026224546259109212, 7.13395138051234912277188883841, 7.948054728561044771508459080490, 8.594552209368300032470014171688, 9.072160984887336108171307850338, 9.367518156617666493058351888766
