| L(s) = 1 | + 3-s − 5-s − 3·11-s − 8·13-s − 15-s − 4·19-s − 8·23-s + 5·25-s − 27-s − 6·29-s − 5·31-s − 3·33-s − 8·37-s − 8·39-s − 16·41-s + 12·43-s + 10·47-s − 9·53-s + 3·55-s − 4·57-s − 5·59-s − 10·61-s + 8·65-s − 6·67-s − 8·69-s + 20·71-s + 2·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.904·11-s − 2.21·13-s − 0.258·15-s − 0.917·19-s − 1.66·23-s + 25-s − 0.192·27-s − 1.11·29-s − 0.898·31-s − 0.522·33-s − 1.31·37-s − 1.28·39-s − 2.49·41-s + 1.82·43-s + 1.45·47-s − 1.23·53-s + 0.404·55-s − 0.529·57-s − 0.650·59-s − 1.28·61-s + 0.992·65-s − 0.733·67-s − 0.963·69-s + 2.37·71-s + 0.234·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1382976 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1382976 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3257524975\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3257524975\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 4 T - 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 8 T + 41 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 5 T - 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 8 T + 27 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 10 T + 53 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 9 T + 28 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 5 T - 34 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 10 T + 39 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 6 T - 31 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 2 T - 69 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 11 T + 42 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 18 T + 235 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 17 T + p T^{2} )^{2} \) |
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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
−10.06749897276807562380995735197, −9.505170649448269102904729746493, −9.221896871445982344606302617137, −8.587069911167798286202816501688, −8.474481515614371014088021203809, −7.77907743947412081275116844027, −7.63210037556117839773482525625, −7.09843605574030995706815060634, −7.00510921595633088003778287705, −6.11348381261918118207068011169, −5.79261903810868081611377339783, −5.11911815814601683872115214767, −4.92146307206248120599559016505, −4.30124119034943422008497122490, −3.88446471181808837278549144498, −3.18454238418686780597341920605, −2.81596252233036626975001297059, −1.97319637116671645053221329688, −1.97285711791520985186798182518, −0.21726418110364784231929934538,
0.21726418110364784231929934538, 1.97285711791520985186798182518, 1.97319637116671645053221329688, 2.81596252233036626975001297059, 3.18454238418686780597341920605, 3.88446471181808837278549144498, 4.30124119034943422008497122490, 4.92146307206248120599559016505, 5.11911815814601683872115214767, 5.79261903810868081611377339783, 6.11348381261918118207068011169, 7.00510921595633088003778287705, 7.09843605574030995706815060634, 7.63210037556117839773482525625, 7.77907743947412081275116844027, 8.474481515614371014088021203809, 8.587069911167798286202816501688, 9.221896871445982344606302617137, 9.505170649448269102904729746493, 10.06749897276807562380995735197
