| L(s) = 1 | + 7·2-s + 6·3-s + 25·4-s + 42·6-s + 14·7-s + 63·8-s + 27·9-s − 26·11-s + 150·12-s − 14·13-s + 98·14-s + 169·16-s − 16·17-s + 189·18-s + 174·19-s + 84·21-s − 182·22-s − 184·23-s + 378·24-s − 98·26-s + 108·27-s + 350·28-s − 32·29-s + 330·31-s + 623·32-s − 156·33-s − 112·34-s + ⋯ |
| L(s) = 1 | + 2.47·2-s + 1.15·3-s + 25/8·4-s + 2.85·6-s + 0.755·7-s + 2.78·8-s + 9-s − 0.712·11-s + 3.60·12-s − 0.298·13-s + 1.87·14-s + 2.64·16-s − 0.228·17-s + 2.47·18-s + 2.10·19-s + 0.872·21-s − 1.76·22-s − 1.66·23-s + 3.21·24-s − 0.739·26-s + 0.769·27-s + 2.36·28-s − 0.204·29-s + 1.91·31-s + 3.44·32-s − 0.822·33-s − 0.564·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 275625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275625 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(23.48066849\) |
| \(L(\frac12)\) |
\(\approx\) |
\(23.48066849\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 - 7 T + 3 p^{3} T^{2} - 7 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 26 T + 2406 T^{2} + 26 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 14 T + 2386 T^{2} + 14 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 16 T + 6558 T^{2} + 16 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 174 T + 18414 T^{2} - 174 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 8 p T + 19470 T^{2} + 8 p^{4} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 32 T + 19046 T^{2} + 32 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 330 T + 85430 T^{2} - 330 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 132 T + 103214 T^{2} - 132 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 200 T + 64542 T^{2} - 200 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 364 T + 172486 T^{2} + 364 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 292 T + 179390 T^{2} + 292 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 34 T + 103410 T^{2} + 34 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 364 T + 438374 T^{2} - 364 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 792 T + 340886 T^{2} - 792 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 788 T + 753430 T^{2} - 788 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 454 T + 304526 T^{2} - 454 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 778 T + 794698 T^{2} + 778 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 408 T + 994782 T^{2} - 408 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 1136 T + 1169990 T^{2} + 1136 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 36 T - 842170 T^{2} - 36 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 498 T + 1796754 T^{2} - 498 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−10.88032428014384952711646082297, −10.05816715771281676663580641144, −9.874111266365161741696090583333, −9.694076827192854352773019618574, −8.632977594538730432476033481034, −8.228901147222553152392986686454, −7.87213968152110920507503101515, −7.58964859577051820305915372654, −6.86972513499789386662028036342, −6.44720590118718910214830258451, −5.65292489508851420025133549798, −5.43710430707735670729802763654, −4.80550040150965623487527441687, −4.58985579663825625763032156798, −3.81726383068183278142279404957, −3.66118016233460254266342254073, −2.71509191767347489398220919773, −2.68923710578551387981650502528, −1.77761523432823239856525703269, −0.948285170506407927181090634119,
0.948285170506407927181090634119, 1.77761523432823239856525703269, 2.68923710578551387981650502528, 2.71509191767347489398220919773, 3.66118016233460254266342254073, 3.81726383068183278142279404957, 4.58985579663825625763032156798, 4.80550040150965623487527441687, 5.43710430707735670729802763654, 5.65292489508851420025133549798, 6.44720590118718910214830258451, 6.86972513499789386662028036342, 7.58964859577051820305915372654, 7.87213968152110920507503101515, 8.228901147222553152392986686454, 8.632977594538730432476033481034, 9.694076827192854352773019618574, 9.874111266365161741696090583333, 10.05816715771281676663580641144, 10.88032428014384952711646082297
