L(s) = 1 | − 3-s − 2·5-s − 7-s + 9-s + 4·11-s − 2·13-s + 2·15-s − 2·17-s − 4·19-s + 21-s − 25-s − 27-s − 8·31-s − 4·33-s + 2·35-s + 10·37-s + 2·39-s + 6·41-s + 12·43-s − 2·45-s − 8·47-s + 49-s + 2·51-s + 6·53-s − 8·55-s + 4·57-s − 12·59-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.894·5-s − 0.377·7-s + 1/3·9-s + 1.20·11-s − 0.554·13-s + 0.516·15-s − 0.485·17-s − 0.917·19-s + 0.218·21-s − 1/5·25-s − 0.192·27-s − 1.43·31-s − 0.696·33-s + 0.338·35-s + 1.64·37-s + 0.320·39-s + 0.937·41-s + 1.82·43-s − 0.298·45-s − 1.16·47-s + 1/7·49-s + 0.280·51-s + 0.824·53-s − 1.07·55-s + 0.529·57-s − 1.56·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 282576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 282576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.500596409\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.500596409\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 29 | \( 1 \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.68816608773369, −12.31149448604461, −11.79067144474547, −11.29887140143801, −11.05171261218331, −10.68671214064600, −9.808010704837182, −9.602026952235684, −9.085446055137037, −8.643324828416886, −7.934670149112955, −7.609218166265166, −7.093276117964553, −6.548549115812716, −6.266348555601095, −5.666020267044521, −5.119966671112702, −4.330705363220766, −4.154162166283692, −3.754555235540264, −3.023855888554962, −2.275002211267248, −1.822979914065794, −0.8236759671951234, −0.4559025684016048,
0.4559025684016048, 0.8236759671951234, 1.822979914065794, 2.275002211267248, 3.023855888554962, 3.754555235540264, 4.154162166283692, 4.330705363220766, 5.119966671112702, 5.666020267044521, 6.266348555601095, 6.548549115812716, 7.093276117964553, 7.609218166265166, 7.934670149112955, 8.643324828416886, 9.085446055137037, 9.602026952235684, 9.808010704837182, 10.68671214064600, 11.05171261218331, 11.29887140143801, 11.79067144474547, 12.31149448604461, 12.68816608773369