L(s) = 1 | + 3-s + 2·5-s + 7-s − 2·9-s − 13-s + 2·15-s − 8·17-s − 2·19-s + 21-s − 25-s − 5·27-s + 7·29-s + 7·31-s + 2·35-s + 2·37-s − 39-s − 11·41-s + 4·43-s − 4·45-s + 3·47-s + 49-s − 8·51-s − 6·53-s − 2·57-s − 4·59-s − 2·63-s − 2·65-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.894·5-s + 0.377·7-s − 2/3·9-s − 0.277·13-s + 0.516·15-s − 1.94·17-s − 0.458·19-s + 0.218·21-s − 1/5·25-s − 0.962·27-s + 1.29·29-s + 1.25·31-s + 0.338·35-s + 0.328·37-s − 0.160·39-s − 1.71·41-s + 0.609·43-s − 0.596·45-s + 0.437·47-s + 1/7·49-s − 1.12·51-s − 0.824·53-s − 0.264·57-s − 0.520·59-s − 0.251·63-s − 0.248·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 236992 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 236992 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.210387169\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.210387169\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 \) |
good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 8 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 7 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 11 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 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
−13.11441486289683, −12.41094024243678, −12.01481953893820, −11.48605825798778, −10.95438714736795, −10.67121165525678, −9.994582311235239, −9.614709998789336, −9.150873520352635, −8.600345519846478, −8.348903208497508, −7.921001643928684, −7.147526848766758, −6.609112590244070, −6.277802691849391, −5.817276198553448, −5.015181971412739, −4.775319002236476, −4.153087200861118, −3.531508113834355, −2.698951672652658, −2.481995646987370, −1.978418229033298, −1.319111012412862, −0.3698146962133450,
0.3698146962133450, 1.319111012412862, 1.978418229033298, 2.481995646987370, 2.698951672652658, 3.531508113834355, 4.153087200861118, 4.775319002236476, 5.015181971412739, 5.817276198553448, 6.277802691849391, 6.609112590244070, 7.147526848766758, 7.921001643928684, 8.348903208497508, 8.600345519846478, 9.150873520352635, 9.614709998789336, 9.994582311235239, 10.67121165525678, 10.95438714736795, 11.48605825798778, 12.01481953893820, 12.41094024243678, 13.11441486289683