| L(s) = 1 | + 2·2-s − 2·3-s + 3·4-s − 4·6-s − 2·7-s + 4·8-s − 3·9-s + 4·11-s − 6·12-s + 2·13-s − 4·14-s + 5·16-s − 6·17-s − 6·18-s + 6·19-s + 4·21-s + 8·22-s − 8·23-s − 8·24-s − 5·25-s + 4·26-s + 14·27-s − 6·28-s − 2·29-s − 6·31-s + 6·32-s − 8·33-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1.15·3-s + 3/2·4-s − 1.63·6-s − 0.755·7-s + 1.41·8-s − 9-s + 1.20·11-s − 1.73·12-s + 0.554·13-s − 1.06·14-s + 5/4·16-s − 1.45·17-s − 1.41·18-s + 1.37·19-s + 0.872·21-s + 1.70·22-s − 1.66·23-s − 1.63·24-s − 25-s + 0.784·26-s + 2.69·27-s − 1.13·28-s − 0.371·29-s − 1.07·31-s + 1.06·32-s − 1.39·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36409156 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36409156 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| 431 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 4 T + 21 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 2 T + 22 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 6 T + 38 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 + 8 T + 57 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 2 T + 39 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 6 T + 26 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 4 T - 2 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 4 T + 10 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 61 | $C_4$ | \( 1 + 16 T + 166 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 6 T + 98 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 6 T + 106 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 10 T + 166 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 + 22 T + 282 T^{2} + 22 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 4 T + 162 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 4 T + 153 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
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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
−7.60509530807806939931715273721, −7.44482984300991408381424594777, −6.87559956257109589561844998795, −6.74782560390204799926369824700, −6.10383951980875863985090685867, −6.08056118779065143879496890316, −5.78563298808023909208452403107, −5.62396144851045154871533048667, −5.11921809614326345200059999305, −4.56660702233775397931602344751, −4.24152183511787127457770072620, −4.02559724548440029065740793764, −3.34472752410415697032471886862, −3.31637797868113239237007190613, −2.62127165147352686902719652771, −2.33264220122667763813456872787, −1.55947115768518815396984644050, −1.22632648015447078784130530159, 0, 0,
1.22632648015447078784130530159, 1.55947115768518815396984644050, 2.33264220122667763813456872787, 2.62127165147352686902719652771, 3.31637797868113239237007190613, 3.34472752410415697032471886862, 4.02559724548440029065740793764, 4.24152183511787127457770072620, 4.56660702233775397931602344751, 5.11921809614326345200059999305, 5.62396144851045154871533048667, 5.78563298808023909208452403107, 6.08056118779065143879496890316, 6.10383951980875863985090685867, 6.74782560390204799926369824700, 6.87559956257109589561844998795, 7.44482984300991408381424594777, 7.60509530807806939931715273721
