L(s) = 1 | + 3-s − 7-s − 2·9-s + 5·11-s − 5·13-s + 6·19-s − 21-s − 23-s − 5·25-s − 5·27-s + 6·29-s − 8·31-s + 5·33-s − 4·37-s − 5·39-s + 5·43-s + 9·47-s − 6·49-s + 4·53-s + 6·57-s + 12·59-s + 7·61-s + 2·63-s + 9·67-s − 69-s + 2·71-s + 14·73-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.377·7-s − 2/3·9-s + 1.50·11-s − 1.38·13-s + 1.37·19-s − 0.218·21-s − 0.208·23-s − 25-s − 0.962·27-s + 1.11·29-s − 1.43·31-s + 0.870·33-s − 0.657·37-s − 0.800·39-s + 0.762·43-s + 1.31·47-s − 6/7·49-s + 0.549·53-s + 0.794·57-s + 1.56·59-s + 0.896·61-s + 0.251·63-s + 1.09·67-s − 0.120·69-s + 0.237·71-s + 1.63·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.138993437\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.138993437\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 503 | \( 1 + T \) |
good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 - 9 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−7.78829012253991157852687831328, −7.17531312496346349486605978729, −6.59270248615663449346997814544, −5.66200039329104114395664075980, −5.16242797539331303636501779213, −4.01498622972097185908426134012, −3.56430121238500625139115828258, −2.66996335907092120359538761136, −1.94147567356299341211219717749, −0.68874609506443717062277209605,
0.68874609506443717062277209605, 1.94147567356299341211219717749, 2.66996335907092120359538761136, 3.56430121238500625139115828258, 4.01498622972097185908426134012, 5.16242797539331303636501779213, 5.66200039329104114395664075980, 6.59270248615663449346997814544, 7.17531312496346349486605978729, 7.78829012253991157852687831328