| L(s) = 1 | + 3·3-s + 5-s + 6·9-s + 6·11-s + 3·15-s + 4·23-s − 4·25-s + 9·27-s + 13·31-s + 18·33-s − 2·37-s + 6·45-s + 4·47-s − 6·49-s + 5·53-s + 6·55-s − 10·59-s − 23·67-s + 12·69-s + 17·71-s − 12·75-s + 9·81-s + 7·89-s + 39·93-s + 36·99-s − 103-s − 6·111-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 0.447·5-s + 2·9-s + 1.80·11-s + 0.774·15-s + 0.834·23-s − 4/5·25-s + 1.73·27-s + 2.33·31-s + 3.13·33-s − 0.328·37-s + 0.894·45-s + 0.583·47-s − 6/7·49-s + 0.686·53-s + 0.809·55-s − 1.30·59-s − 2.80·67-s + 1.44·69-s + 2.01·71-s − 1.38·75-s + 81-s + 0.741·89-s + 4.04·93-s + 3.61·99-s − 0.0985·103-s − 0.569·111-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1742400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1742400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.191074436\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.191074436\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 5 | $C_2$ | \( 1 - T + p T^{2} \) |
| 11 | $C_2$ | \( 1 - 6 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 24 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 105 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 11 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 125 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 103 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
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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
−8.005547990151123980806648723671, −7.44613934147934016835887366907, −7.00881869093392385911889825173, −6.67217626490903331028365662485, −6.16095780661827101539146702850, −5.90947905716997979401654183871, −4.97963578657367743477052941603, −4.63088973892072176736621690664, −4.10053527921656446400602215032, −3.75129072676887626651703068040, −3.16070782693756096454411779287, −2.78353860540573015518026042666, −2.14700936271689844060665848980, −1.54268219585144971317205380181, −1.04852898241242524607739096269,
1.04852898241242524607739096269, 1.54268219585144971317205380181, 2.14700936271689844060665848980, 2.78353860540573015518026042666, 3.16070782693756096454411779287, 3.75129072676887626651703068040, 4.10053527921656446400602215032, 4.63088973892072176736621690664, 4.97963578657367743477052941603, 5.90947905716997979401654183871, 6.16095780661827101539146702850, 6.67217626490903331028365662485, 7.00881869093392385911889825173, 7.44613934147934016835887366907, 8.005547990151123980806648723671
