| L(s) = 1 | + 4·2-s + 8·4-s + 4·5-s + 32·8-s + 16·10-s + 62·11-s + 124·13-s + 128·16-s − 84·17-s + 100·19-s + 32·20-s + 248·22-s − 42·23-s + 125·25-s + 496·26-s + 20·29-s − 48·31-s + 256·32-s − 336·34-s + 246·37-s + 400·38-s + 128·40-s − 496·41-s + 136·43-s + 496·44-s − 168·46-s − 324·47-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s + 0.357·5-s + 1.41·8-s + 0.505·10-s + 1.69·11-s + 2.64·13-s + 2·16-s − 1.19·17-s + 1.20·19-s + 0.357·20-s + 2.40·22-s − 0.380·23-s + 25-s + 3.74·26-s + 0.128·29-s − 0.278·31-s + 1.41·32-s − 1.69·34-s + 1.09·37-s + 1.70·38-s + 0.505·40-s − 1.88·41-s + 0.482·43-s + 1.69·44-s − 0.538·46-s − 1.00·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(11.56114806\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.56114806\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - p^{2} T + p^{3} T^{2} - p^{5} T^{3} + p^{6} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 4 T - 109 T^{2} - 4 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 62 T + 2513 T^{2} - 62 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 62 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 84 T + 2143 T^{2} + 84 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 100 T + 3141 T^{2} - 100 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 42 T - 10403 T^{2} + 42 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 48 T - 27487 T^{2} + 48 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 246 T + 9863 T^{2} - 246 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 248 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 68 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 324 T + 1153 T^{2} + 324 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 258 T - 82313 T^{2} - 258 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 120 T - 190979 T^{2} + 120 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 622 T + 159903 T^{2} - 622 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 904 T + 516453 T^{2} + 904 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 678 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 642 T + 23147 T^{2} + 642 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 740 T + 54561 T^{2} + 740 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 468 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 200 T - 664969 T^{2} + 200 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 1266 T + p^{3} T^{2} )^{2} \) |
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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
−10.97243618838578852741091962799, −10.79140244527796179132987238589, −10.17374828020073240013522114154, −9.533804085030187635325435411250, −9.167653644842767042435496421138, −8.483697254092750268115934490116, −8.408040075433781663016324565477, −7.58581335285119630267708267129, −6.86993161531219033184036837264, −6.62719138379382806629512412425, −6.18687543893505468745118794139, −5.67499072268838903026670057578, −5.17827915420782659295821722332, −4.41862198501957200140632077060, −4.18982102974614174638649916255, −3.40961292006127990460344089987, −3.39893247367747519061621449493, −2.12097258348363533111229685037, −1.34367975921383036261734921342, −1.07237533859292919610992333894,
1.07237533859292919610992333894, 1.34367975921383036261734921342, 2.12097258348363533111229685037, 3.39893247367747519061621449493, 3.40961292006127990460344089987, 4.18982102974614174638649916255, 4.41862198501957200140632077060, 5.17827915420782659295821722332, 5.67499072268838903026670057578, 6.18687543893505468745118794139, 6.62719138379382806629512412425, 6.86993161531219033184036837264, 7.58581335285119630267708267129, 8.408040075433781663016324565477, 8.483697254092750268115934490116, 9.167653644842767042435496421138, 9.533804085030187635325435411250, 10.17374828020073240013522114154, 10.79140244527796179132987238589, 10.97243618838578852741091962799
