| L(s) = 1 | + 2·5-s + 4·11-s + 8·17-s + 8·19-s + 12·23-s + 3·25-s + 4·29-s − 4·37-s + 4·41-s − 8·43-s + 4·47-s + 12·53-s + 8·55-s − 8·67-s + 4·71-s − 8·73-s − 20·79-s + 4·83-s + 16·85-s + 4·89-s + 16·95-s + 8·97-s − 12·101-s + 4·107-s − 20·109-s + 4·113-s + 24·115-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 1.20·11-s + 1.94·17-s + 1.83·19-s + 2.50·23-s + 3/5·25-s + 0.742·29-s − 0.657·37-s + 0.624·41-s − 1.21·43-s + 0.583·47-s + 1.64·53-s + 1.07·55-s − 0.977·67-s + 0.474·71-s − 0.936·73-s − 2.25·79-s + 0.439·83-s + 1.73·85-s + 0.423·89-s + 1.64·95-s + 0.812·97-s − 1.19·101-s + 0.386·107-s − 1.91·109-s + 0.376·113-s + 2.23·115-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 77792400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77792400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.459657707\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.459657707\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | | \( 1 \) |
| good | 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 8 T + 42 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 8 T + 52 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 12 T + 80 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 44 T^{2} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 4 T + 26 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 12 T + 140 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 110 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 24 T^{2} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 8 T + 142 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 4 T + 74 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 8 T + 90 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 - 4 T + 162 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 4 T + 174 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
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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
−7.74712033516730431473372574603, −7.52846233826395426255973407506, −7.16713685485766645048430074759, −7.01383579709068447954719237792, −6.44779118097064378341323607403, −6.38741445795664512481647015119, −5.65849528119130049581914814029, −5.54247770818693467371308501795, −5.13766579129356746648144487245, −5.13380425038798007123967452937, −4.28660602838065974899252470912, −4.19882927046724117591422500221, −3.36959714532770377512894494684, −3.34922217339912349409870031811, −2.79825987665496331429859113815, −2.71440068843703063953423943298, −1.68417472475699939266580525028, −1.51308334383486734356373123213, −0.952848583396612255265534872428, −0.798431694788880264209776077711,
0.798431694788880264209776077711, 0.952848583396612255265534872428, 1.51308334383486734356373123213, 1.68417472475699939266580525028, 2.71440068843703063953423943298, 2.79825987665496331429859113815, 3.34922217339912349409870031811, 3.36959714532770377512894494684, 4.19882927046724117591422500221, 4.28660602838065974899252470912, 5.13380425038798007123967452937, 5.13766579129356746648144487245, 5.54247770818693467371308501795, 5.65849528119130049581914814029, 6.38741445795664512481647015119, 6.44779118097064378341323607403, 7.01383579709068447954719237792, 7.16713685485766645048430074759, 7.52846233826395426255973407506, 7.74712033516730431473372574603
