L(s) = 1 | + 2-s + 4-s − 3·5-s + 7-s + 8-s − 3·10-s + 2·13-s + 14-s + 16-s − 17-s + 7·19-s − 3·20-s − 23-s + 4·25-s + 2·26-s + 28-s + 8·29-s − 9·31-s + 32-s − 34-s − 3·35-s − 5·37-s + 7·38-s − 3·40-s + 2·41-s − 8·43-s − 46-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s − 1.34·5-s + 0.377·7-s + 0.353·8-s − 0.948·10-s + 0.554·13-s + 0.267·14-s + 1/4·16-s − 0.242·17-s + 1.60·19-s − 0.670·20-s − 0.208·23-s + 4/5·25-s + 0.392·26-s + 0.188·28-s + 1.48·29-s − 1.61·31-s + 0.176·32-s − 0.171·34-s − 0.507·35-s − 0.821·37-s + 1.13·38-s − 0.474·40-s + 0.312·41-s − 1.21·43-s − 0.147·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 259182 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 259182 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.270467823\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.270467823\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 11 | \( 1 \) |
| 17 | \( 1 + T \) |
good | 5 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + 9 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 11 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 3 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 + 7 T + p T^{2} \) |
| 97 | \( 1 + 9 T + p T^{2} \) |
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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
−12.61383553582570, −12.26217275280497, −12.03067243058453, −11.47167967327613, −11.11468852936730, −10.76751416812274, −10.19930818969256, −9.565573632763120, −9.017934980519969, −8.537279679075016, −7.923437366138696, −7.704710015177908, −7.222429513523067, −6.645941436803022, −6.217521895047461, −5.493764948663509, −5.022828830144656, −4.681387577758908, −3.984932115362002, −3.618627370426701, −3.160433530081627, −2.652098599138663, −1.658888015450459, −1.296319491168591, −0.3664979693578568,
0.3664979693578568, 1.296319491168591, 1.658888015450459, 2.652098599138663, 3.160433530081627, 3.618627370426701, 3.984932115362002, 4.681387577758908, 5.022828830144656, 5.493764948663509, 6.217521895047461, 6.645941436803022, 7.222429513523067, 7.704710015177908, 7.923437366138696, 8.537279679075016, 9.017934980519969, 9.565573632763120, 10.19930818969256, 10.76751416812274, 11.11468852936730, 11.47167967327613, 12.03067243058453, 12.26217275280497, 12.61383553582570