| L(s) = 1 | + 3.11·3-s − 5-s − 0.964·7-s + 6.71·9-s − 2.75·11-s + 3.87·13-s − 3.11·15-s + 2.11·17-s + 5.82·19-s − 3.00·21-s − 2.04·23-s + 25-s + 11.5·27-s + 29-s + 2.96·31-s − 8.58·33-s + 0.964·35-s − 4.75·37-s + 12.0·39-s + 11.3·41-s + 3.27·43-s − 6.71·45-s + 0.405·47-s − 6.07·49-s + 6.58·51-s − 6.98·53-s + 2.75·55-s + ⋯ |
| L(s) = 1 | + 1.79·3-s − 0.447·5-s − 0.364·7-s + 2.23·9-s − 0.830·11-s + 1.07·13-s − 0.804·15-s + 0.512·17-s + 1.33·19-s − 0.655·21-s − 0.425·23-s + 0.200·25-s + 2.23·27-s + 0.185·29-s + 0.532·31-s − 1.49·33-s + 0.162·35-s − 0.781·37-s + 1.93·39-s + 1.76·41-s + 0.498·43-s − 1.00·45-s + 0.0591·47-s − 0.867·49-s + 0.921·51-s − 0.960·53-s + 0.371·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.154411251\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.154411251\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 - 3.11T + 3T^{2} \) |
| 7 | \( 1 + 0.964T + 7T^{2} \) |
| 11 | \( 1 + 2.75T + 11T^{2} \) |
| 13 | \( 1 - 3.87T + 13T^{2} \) |
| 17 | \( 1 - 2.11T + 17T^{2} \) |
| 19 | \( 1 - 5.82T + 19T^{2} \) |
| 23 | \( 1 + 2.04T + 23T^{2} \) |
| 31 | \( 1 - 2.96T + 31T^{2} \) |
| 37 | \( 1 + 4.75T + 37T^{2} \) |
| 41 | \( 1 - 11.3T + 41T^{2} \) |
| 43 | \( 1 - 3.27T + 43T^{2} \) |
| 47 | \( 1 - 0.405T + 47T^{2} \) |
| 53 | \( 1 + 6.98T + 53T^{2} \) |
| 59 | \( 1 + 0.364T + 59T^{2} \) |
| 61 | \( 1 + 12.1T + 61T^{2} \) |
| 67 | \( 1 + 1.48T + 67T^{2} \) |
| 71 | \( 1 - 14.5T + 71T^{2} \) |
| 73 | \( 1 + 2.16T + 73T^{2} \) |
| 79 | \( 1 - 14.1T + 79T^{2} \) |
| 83 | \( 1 + 11.3T + 83T^{2} \) |
| 89 | \( 1 - 6.08T + 89T^{2} \) |
| 97 | \( 1 + 2.31T + 97T^{2} \) |
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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
−7.914449463449915514621696128672, −7.37429492930043605401465481194, −6.52764256821701076314579669704, −5.66835177280226087181888166645, −4.71171534632721233157341114134, −3.94100656820025680503605199694, −3.23842939970030489528123864160, −2.92331481402841027094978594330, −1.90277622373425806244971794849, −0.926486399825057779221988889953,
0.926486399825057779221988889953, 1.90277622373425806244971794849, 2.92331481402841027094978594330, 3.23842939970030489528123864160, 3.94100656820025680503605199694, 4.71171534632721233157341114134, 5.66835177280226087181888166645, 6.52764256821701076314579669704, 7.37429492930043605401465481194, 7.914449463449915514621696128672
