| L(s) = 1 | − 6·3-s − 10·5-s − 12·7-s + 27·9-s + 24·11-s + 80·13-s + 60·15-s + 40·17-s + 36·19-s + 72·21-s − 108·23-s + 75·25-s − 108·27-s − 108·29-s − 516·31-s − 144·33-s + 120·35-s + 448·37-s − 480·39-s − 212·41-s − 456·43-s − 270·45-s − 756·47-s − 414·49-s − 240·51-s + 252·53-s − 240·55-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.894·5-s − 0.647·7-s + 9-s + 0.657·11-s + 1.70·13-s + 1.03·15-s + 0.570·17-s + 0.434·19-s + 0.748·21-s − 0.979·23-s + 3/5·25-s − 0.769·27-s − 0.691·29-s − 2.98·31-s − 0.759·33-s + 0.579·35-s + 1.99·37-s − 1.97·39-s − 0.807·41-s − 1.61·43-s − 0.894·45-s − 2.34·47-s − 1.20·49-s − 0.658·51-s + 0.653·53-s − 0.588·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230400 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 5 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 + 12 T + 558 T^{2} + 12 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 24 T + 182 T^{2} - 24 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 80 T + 4518 T^{2} - 80 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 40 T - 3058 T^{2} - 40 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 36 T + 758 T^{2} - 36 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 108 T + 27086 T^{2} + 108 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 108 T + 28078 T^{2} + 108 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 516 T + 122046 T^{2} + 516 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 448 T + 138198 T^{2} - 448 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 212 T + 1478 T^{2} + 212 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 456 T + 187382 T^{2} + 456 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 756 T + 322814 T^{2} + 756 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 252 T + 24334 T^{2} - 252 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 792 T + 564950 T^{2} + 792 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 164 T + 454782 T^{2} - 164 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 216 T + 295686 T^{2} - 216 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 1056 T + 984110 T^{2} + 1056 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 36 T + 772454 T^{2} + 36 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 2028 T + 1955070 T^{2} + 2028 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 360 T + 1008038 T^{2} + 360 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 1588 T + 1892774 T^{2} + 1588 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 1540 T + 1568070 T^{2} - 1540 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−10.37607398435961941555825642989, −10.00829178048293484955686210950, −9.458902466291272623195879919828, −9.202320977326638206761002164845, −8.336813857688250112935537590976, −8.232601568852549157687448305181, −7.44206678813944824754395868209, −7.16737064947454009669521405577, −6.41180843857549426894385044141, −6.31648345799553570648203287275, −5.56627186879273170179887899956, −5.38930254185735005843191424292, −4.38718440621161168133058639668, −4.08596190680334104656907728081, −3.40508899725484278063292756887, −3.19447690847969023132534999423, −1.51906223815079821197587200245, −1.45836112621855294670865571343, 0, 0,
1.45836112621855294670865571343, 1.51906223815079821197587200245, 3.19447690847969023132534999423, 3.40508899725484278063292756887, 4.08596190680334104656907728081, 4.38718440621161168133058639668, 5.38930254185735005843191424292, 5.56627186879273170179887899956, 6.31648345799553570648203287275, 6.41180843857549426894385044141, 7.16737064947454009669521405577, 7.44206678813944824754395868209, 8.232601568852549157687448305181, 8.336813857688250112935537590976, 9.202320977326638206761002164845, 9.458902466291272623195879919828, 10.00829178048293484955686210950, 10.37607398435961941555825642989
