| L(s) = 1 | − 3·5-s + 7-s + 5·25-s − 3·35-s + 2·37-s + 3·41-s − 43-s + 3·47-s + 3·59-s − 2·67-s + 79-s − 3·83-s + 2·109-s + 121-s − 6·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s + 173-s + 5·175-s + ⋯ |
| L(s) = 1 | − 3·5-s + 7-s + 5·25-s − 3·35-s + 2·37-s + 3·41-s − 43-s + 3·47-s + 3·59-s − 2·67-s + 79-s − 3·83-s + 2·109-s + 121-s − 6·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s + 173-s + 5·175-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5143824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5143824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7685755953\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7685755953\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 5 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 23 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 29 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 41 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 47 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 59 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 83 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 97 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
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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
−9.078949149296364721844578620518, −8.898364953839260081344464142834, −8.535340671225328278545687727690, −8.082530694953781923856075795120, −7.945544919764413663153684362676, −7.41449408425566771646105905193, −7.21506058530878837167490205310, −7.21036665033261675684970223857, −6.16629090741261463058074659633, −6.00753273252048260949815062299, −5.32008612071121046338432533705, −4.85775703499821089842068727015, −4.28584064779381969175476233229, −4.25458126245158779272629838271, −3.88614105807745036424933551367, −3.41615097084552867765963548166, −2.55028956964752775882841536748, −2.54753201292880288124664834330, −1.22125701575781913111418434407, −0.71508009158832481173920418565,
0.71508009158832481173920418565, 1.22125701575781913111418434407, 2.54753201292880288124664834330, 2.55028956964752775882841536748, 3.41615097084552867765963548166, 3.88614105807745036424933551367, 4.25458126245158779272629838271, 4.28584064779381969175476233229, 4.85775703499821089842068727015, 5.32008612071121046338432533705, 6.00753273252048260949815062299, 6.16629090741261463058074659633, 7.21036665033261675684970223857, 7.21506058530878837167490205310, 7.41449408425566771646105905193, 7.945544919764413663153684362676, 8.082530694953781923856075795120, 8.535340671225328278545687727690, 8.898364953839260081344464142834, 9.078949149296364721844578620518
