| L(s) = 1 | − 2·2-s + 3·4-s − 4·8-s − 9-s − 4·13-s + 5·16-s + 2·17-s + 2·18-s + 8·26-s − 6·32-s − 4·34-s − 3·36-s − 24·43-s + 16·47-s + 10·49-s − 12·52-s − 20·53-s − 16·59-s + 7·64-s + 6·68-s + 4·72-s + 81-s + 48·86-s − 12·89-s − 32·94-s − 20·98-s − 20·101-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s − 1.41·8-s − 1/3·9-s − 1.10·13-s + 5/4·16-s + 0.485·17-s + 0.471·18-s + 1.56·26-s − 1.06·32-s − 0.685·34-s − 1/2·36-s − 3.65·43-s + 2.33·47-s + 10/7·49-s − 1.66·52-s − 2.74·53-s − 2.08·59-s + 7/8·64-s + 0.727·68-s + 0.471·72-s + 1/9·81-s + 5.17·86-s − 1.27·89-s − 3.30·94-s − 2.02·98-s − 1.99·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6502500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6502500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2020257480\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2020257480\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 17 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 154 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
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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
−9.380618844291277153183250241710, −8.597079573993857743578352716006, −8.387528608495091167256175398592, −8.013913585552573613125966666554, −7.70318415902264446702163052909, −7.14712601657997585869460359875, −7.06874855228999589837387557460, −6.52842705798941927162555154800, −6.16837785140986143027572063896, −5.54515194313369805197706519314, −5.45595128004983152343399377295, −4.61376489533499235176858762002, −4.56332895986714741291859705285, −3.54005281454644461656787721003, −3.36077063950459961028271732870, −2.63148581679277215716358367387, −2.44171596982709289330493380803, −1.57981208237453119001531501400, −1.30713764913141909737749471925, −0.19176141319988473126438787106,
0.19176141319988473126438787106, 1.30713764913141909737749471925, 1.57981208237453119001531501400, 2.44171596982709289330493380803, 2.63148581679277215716358367387, 3.36077063950459961028271732870, 3.54005281454644461656787721003, 4.56332895986714741291859705285, 4.61376489533499235176858762002, 5.45595128004983152343399377295, 5.54515194313369805197706519314, 6.16837785140986143027572063896, 6.52842705798941927162555154800, 7.06874855228999589837387557460, 7.14712601657997585869460359875, 7.70318415902264446702163052909, 8.013913585552573613125966666554, 8.387528608495091167256175398592, 8.597079573993857743578352716006, 9.380618844291277153183250241710
