L(s) = 1 | − 2-s + 4-s − 5-s − 8-s + 10-s − 3·11-s − 4·13-s + 16-s − 4·17-s − 20-s + 3·22-s + 4·23-s − 4·25-s + 4·26-s − 7·29-s + 7·31-s − 32-s + 4·34-s + 4·37-s + 40-s − 12·41-s − 6·43-s − 3·44-s − 4·46-s + 10·47-s + 4·50-s − 4·52-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.353·8-s + 0.316·10-s − 0.904·11-s − 1.10·13-s + 1/4·16-s − 0.970·17-s − 0.223·20-s + 0.639·22-s + 0.834·23-s − 4/5·25-s + 0.784·26-s − 1.29·29-s + 1.25·31-s − 0.176·32-s + 0.685·34-s + 0.657·37-s + 0.158·40-s − 1.87·41-s − 0.914·43-s − 0.452·44-s − 0.589·46-s + 1.45·47-s + 0.565·50-s − 0.554·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 318402 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 318402 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2157293489\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2157293489\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
| 19 | \( 1 \) |
good | 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 7 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 - 13 T + p T^{2} \) |
| 59 | \( 1 - T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 - 15 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 3 T + p T^{2} \) |
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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
−12.40232176058340, −12.08857065235776, −11.77597226275009, −11.12115149277587, −10.84630944330567, −10.28227535584280, −9.941345733849025, −9.399409091675540, −9.000335569068545, −8.462283478670973, −7.957972354982909, −7.680383684536135, −7.033316219187934, −6.863791273760365, −6.142549258703396, −5.559853866898253, −5.040525060788129, −4.648285546689536, −3.955926775416135, −3.435190014471587, −2.668104483531937, −2.393610031514394, −1.793610632377459, −0.9542025283834192, −0.1554691145362676,
0.1554691145362676, 0.9542025283834192, 1.793610632377459, 2.393610031514394, 2.668104483531937, 3.435190014471587, 3.955926775416135, 4.648285546689536, 5.040525060788129, 5.559853866898253, 6.142549258703396, 6.863791273760365, 7.033316219187934, 7.680383684536135, 7.957972354982909, 8.462283478670973, 9.000335569068545, 9.399409091675540, 9.941345733849025, 10.28227535584280, 10.84630944330567, 11.12115149277587, 11.77597226275009, 12.08857065235776, 12.40232176058340