L(s) = 1 | − 3-s + 4·5-s + 7-s + 9-s + 5·11-s − 2·13-s − 4·15-s + 5·19-s − 21-s + 23-s + 11·25-s − 27-s − 2·29-s − 6·31-s − 5·33-s + 4·35-s + 6·37-s + 2·39-s + 5·41-s − 8·43-s + 4·45-s + 9·47-s + 49-s + 9·53-s + 20·55-s − 5·57-s − 9·59-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1.78·5-s + 0.377·7-s + 1/3·9-s + 1.50·11-s − 0.554·13-s − 1.03·15-s + 1.14·19-s − 0.218·21-s + 0.208·23-s + 11/5·25-s − 0.192·27-s − 0.371·29-s − 1.07·31-s − 0.870·33-s + 0.676·35-s + 0.986·37-s + 0.320·39-s + 0.780·41-s − 1.21·43-s + 0.596·45-s + 1.31·47-s + 1/7·49-s + 1.23·53-s + 2.69·55-s − 0.662·57-s − 1.17·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.097549591\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.097549591\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 23 | \( 1 - T \) |
good | 5 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 18 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−7.63158150513750950337647130419, −7.05817953077553731244250788627, −6.30271385755438495186139887277, −5.82636434969157029969719859702, −5.21400971049369667954496425971, −4.52798276521074803880719468746, −3.53520334126760659677282274992, −2.46725105075483857162906337780, −1.65098046847587676418022247486, −1.00050204410442716469698004603,
1.00050204410442716469698004603, 1.65098046847587676418022247486, 2.46725105075483857162906337780, 3.53520334126760659677282274992, 4.52798276521074803880719468746, 5.21400971049369667954496425971, 5.82636434969157029969719859702, 6.30271385755438495186139887277, 7.05817953077553731244250788627, 7.63158150513750950337647130419