| L(s) = 1 | − 6·3-s + 5·5-s + 27·9-s − 67·11-s + 41·13-s − 30·15-s − 92·17-s − 43·19-s − 148·23-s + 105·25-s − 108·27-s + 77·29-s + 520·31-s + 402·33-s + 7·37-s − 246·39-s − 426·41-s + 107·43-s + 135·45-s − 576·47-s + 552·51-s − 243·53-s − 335·55-s + 258·57-s + 7·59-s + 224·61-s + 205·65-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.447·5-s + 9-s − 1.83·11-s + 0.874·13-s − 0.516·15-s − 1.31·17-s − 0.519·19-s − 1.34·23-s + 0.839·25-s − 0.769·27-s + 0.493·29-s + 3.01·31-s + 2.12·33-s + 0.0311·37-s − 1.01·39-s − 1.62·41-s + 0.379·43-s + 0.447·45-s − 1.78·47-s + 1.51·51-s − 0.629·53-s − 0.821·55-s + 0.599·57-s + 0.0154·59-s + 0.470·61-s + 0.391·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 2 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $D_{4}$ | \( 1 - p T - 16 p T^{2} - p^{4} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 67 T + 3448 T^{2} + 67 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 41 T + 4478 T^{2} - 41 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 92 T + 386 p T^{2} + 92 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 43 T + 11154 T^{2} + 43 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 148 T + 24430 T^{2} + 148 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 77 T + 9574 T^{2} - 77 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 520 T + 125837 T^{2} - 520 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 7 T + 74082 T^{2} - 7 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 426 T + 171106 T^{2} + 426 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 107 T + 86220 T^{2} - 107 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 576 T + 242170 T^{2} + 576 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 243 T + 309490 T^{2} + 243 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 7 T + 200614 T^{2} - 7 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 224 T + 461126 T^{2} - 224 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 687 T + 678832 T^{2} + 687 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 472 T + 637018 T^{2} + 472 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 921 T + 578188 T^{2} + 921 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 526 T + 1033727 T^{2} - 526 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 221 T + 945628 T^{2} - 221 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 774 T + 966562 T^{2} - 774 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 1953 T + 2366992 T^{2} - 1953 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−8.424827353068040964449807627030, −8.122340437210690217727975031924, −7.68956672008341244104272548172, −7.24875083185844534905649454190, −6.65251073256561928793774322768, −6.40154637487857140709595476471, −6.06318922165362113280339716025, −5.96974474896087118668855397558, −5.10109258685485670438334754747, −4.93459891650328400417022885106, −4.59619421857768661326195573090, −4.28370376931732995520035100639, −3.44672059323357481377355697398, −3.07798302522492776500398272227, −2.38979060784836773797738428592, −2.13553312412437488821494559131, −1.40626396190798642345558716108, −0.871860779336508881357669304978, 0, 0,
0.871860779336508881357669304978, 1.40626396190798642345558716108, 2.13553312412437488821494559131, 2.38979060784836773797738428592, 3.07798302522492776500398272227, 3.44672059323357481377355697398, 4.28370376931732995520035100639, 4.59619421857768661326195573090, 4.93459891650328400417022885106, 5.10109258685485670438334754747, 5.96974474896087118668855397558, 6.06318922165362113280339716025, 6.40154637487857140709595476471, 6.65251073256561928793774322768, 7.24875083185844534905649454190, 7.68956672008341244104272548172, 8.122340437210690217727975031924, 8.424827353068040964449807627030
