| L(s) = 1 | + 2-s + 3-s + 3·5-s + 6-s + 5·7-s − 8-s + 3·10-s + 2·11-s + 6·13-s + 5·14-s + 3·15-s − 16-s + 3·17-s + 5·21-s + 2·22-s + 23-s − 24-s + 5·25-s + 6·26-s − 27-s − 16·29-s + 3·30-s − 10·31-s + 2·33-s + 3·34-s + 15·35-s + 2·37-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1.34·5-s + 0.408·6-s + 1.88·7-s − 0.353·8-s + 0.948·10-s + 0.603·11-s + 1.66·13-s + 1.33·14-s + 0.774·15-s − 1/4·16-s + 0.727·17-s + 1.09·21-s + 0.426·22-s + 0.208·23-s − 0.204·24-s + 25-s + 1.17·26-s − 0.192·27-s − 2.97·29-s + 0.547·30-s − 1.79·31-s + 0.348·33-s + 0.514·34-s + 2.53·35-s + 0.328·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 933156 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 933156 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.743525850\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.743525850\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 3 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | $C_2$ | \( 1 - 5 T + p T^{2} \) |
| 23 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 10 T + 69 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 2 T - 33 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 9 T + 34 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 9 T + 28 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 4 T - 43 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 6 T - 25 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 3 T - 58 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 3 T - 64 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 10 T + 11 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 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
−10.17558176096704126609763165759, −9.698978849871421478887554759386, −9.275404821433045568781440017747, −9.067577544070458695879792857461, −8.438386670447348251852927220398, −8.420966649392407149401819269377, −7.58650326605607548864565798909, −7.43381893530918799645010840663, −6.84000641822528066806791789198, −5.99867157075521024151450942029, −5.90851710386689780904072709609, −5.32953523068201392874478541384, −5.28764471442302278185453342159, −4.40271287235392247223877202749, −3.79104943082822696333042728347, −3.77775830799584899352373493507, −2.85845459494656305809856509170, −2.11166071264027231088414276182, −1.64392197025168130289364477576, −1.25888122255994271328450827496,
1.25888122255994271328450827496, 1.64392197025168130289364477576, 2.11166071264027231088414276182, 2.85845459494656305809856509170, 3.77775830799584899352373493507, 3.79104943082822696333042728347, 4.40271287235392247223877202749, 5.28764471442302278185453342159, 5.32953523068201392874478541384, 5.90851710386689780904072709609, 5.99867157075521024151450942029, 6.84000641822528066806791789198, 7.43381893530918799645010840663, 7.58650326605607548864565798909, 8.420966649392407149401819269377, 8.438386670447348251852927220398, 9.067577544070458695879792857461, 9.275404821433045568781440017747, 9.698978849871421478887554759386, 10.17558176096704126609763165759
