| L(s) = 1 | + 4·2-s − 2·3-s + 12·4-s − 8·6-s + 32·8-s − 33·9-s − 26·11-s − 24·12-s + 102·13-s + 80·16-s − 186·17-s − 132·18-s + 36·19-s − 104·22-s + 44·23-s − 64·24-s + 408·26-s + 86·27-s − 46·29-s + 140·31-s + 192·32-s + 52·33-s − 744·34-s − 396·36-s − 132·37-s + 144·38-s − 204·39-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 0.384·3-s + 3/2·4-s − 0.544·6-s + 1.41·8-s − 1.22·9-s − 0.712·11-s − 0.577·12-s + 2.17·13-s + 5/4·16-s − 2.65·17-s − 1.72·18-s + 0.434·19-s − 1.00·22-s + 0.398·23-s − 0.544·24-s + 3.07·26-s + 0.612·27-s − 0.294·29-s + 0.811·31-s + 1.06·32-s + 0.274·33-s − 3.75·34-s − 1.83·36-s − 0.586·37-s + 0.614·38-s − 0.837·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6002500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6002500 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 5 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 3 | $D_{4}$ | \( 1 + 2 T + 37 T^{2} + 2 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 26 T + 2703 T^{2} + 26 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 102 T + 6833 T^{2} - 102 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 186 T + 17897 T^{2} + 186 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 36 T + 13530 T^{2} - 36 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 44 T + 192 p T^{2} - 44 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 46 T + 38939 T^{2} + 46 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 140 T + 38944 T^{2} - 140 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 132 T + 28044 T^{2} + 132 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 132 T + 100148 T^{2} + 132 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 324 T + 181386 T^{2} + 324 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 242 T + 215325 T^{2} + 242 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 208 T + 205512 T^{2} + 208 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 780 T + 529576 T^{2} - 780 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 216 T + 425298 T^{2} - 216 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 256 T + 62452 T^{2} - 256 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 692 T + 813066 T^{2} + 692 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 832 T + 948202 T^{2} + 832 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 1962 T + 1889271 T^{2} + 1962 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 1712 T + 1530198 T^{2} + 1712 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 1960 T + 2217986 T^{2} - 1960 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 102 T + 453465 T^{2} + 102 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−8.378347750140306323389128623434, −8.227474044881204458212793720476, −7.37065464222000800673468057800, −7.10403830452236666712371571804, −6.59334145542790038635706141121, −6.41227455677833102279845012015, −5.88796091524754259117710439961, −5.81686537865356417131273252395, −5.08898573084149269776854451964, −5.02116411775709197716627097859, −4.36808299494908849116400702472, −4.09095780336853338158792904653, −3.43563272009156653785125088937, −3.25109375844706135978504904702, −2.55188912522043031361676748941, −2.37959797363454435325577153628, −1.54204357537045684875380911508, −1.18361913390535218408076586727, 0, 0,
1.18361913390535218408076586727, 1.54204357537045684875380911508, 2.37959797363454435325577153628, 2.55188912522043031361676748941, 3.25109375844706135978504904702, 3.43563272009156653785125088937, 4.09095780336853338158792904653, 4.36808299494908849116400702472, 5.02116411775709197716627097859, 5.08898573084149269776854451964, 5.81686537865356417131273252395, 5.88796091524754259117710439961, 6.41227455677833102279845012015, 6.59334145542790038635706141121, 7.10403830452236666712371571804, 7.37065464222000800673468057800, 8.227474044881204458212793720476, 8.378347750140306323389128623434
