L(s) = 1 | − 2-s + 4-s − 7-s − 8-s + 14-s + 16-s + 4·17-s − 2·19-s + 23-s − 28-s + 2·29-s − 10·31-s − 32-s − 4·34-s + 10·37-s + 2·38-s − 6·41-s − 4·43-s − 46-s + 2·47-s + 49-s − 6·53-s + 56-s − 2·58-s − 4·59-s + 6·61-s + 10·62-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.377·7-s − 0.353·8-s + 0.267·14-s + 1/4·16-s + 0.970·17-s − 0.458·19-s + 0.208·23-s − 0.188·28-s + 0.371·29-s − 1.79·31-s − 0.176·32-s − 0.685·34-s + 1.64·37-s + 0.324·38-s − 0.937·41-s − 0.609·43-s − 0.147·46-s + 0.291·47-s + 1/7·49-s − 0.824·53-s + 0.133·56-s − 0.262·58-s − 0.520·59-s + 0.768·61-s + 1.27·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 10 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−14.50338983076802, −13.85643105436435, −13.29961118508965, −12.69566968203337, −12.45164526464024, −11.71569291520320, −11.34447998991865, −10.69853024527078, −10.32949668859820, −9.754933280022271, −9.273596777717888, −8.898905249654707, −8.153978805538166, −7.756750444288496, −7.296667798516624, −6.489875386839074, −6.312266742666764, −5.443303526164293, −5.075612725191589, −4.159173947836414, −3.562364685903262, −3.012351650037845, −2.286039854980572, −1.602266992908755, −0.8444747269067190, 0,
0.8444747269067190, 1.602266992908755, 2.286039854980572, 3.012351650037845, 3.562364685903262, 4.159173947836414, 5.075612725191589, 5.443303526164293, 6.312266742666764, 6.489875386839074, 7.296667798516624, 7.756750444288496, 8.153978805538166, 8.898905249654707, 9.273596777717888, 9.754933280022271, 10.32949668859820, 10.69853024527078, 11.34447998991865, 11.71569291520320, 12.45164526464024, 12.69566968203337, 13.29961118508965, 13.85643105436435, 14.50338983076802