L(s) = 1 | + 18.4·5-s − 22.4·7-s − 53.6·11-s + 7.15·13-s − 39.6·17-s + 125.·19-s + 99.1·23-s + 214.·25-s − 205.·29-s − 147.·31-s − 413.·35-s + 125.·37-s + 506.·41-s − 413.·43-s − 313.·47-s + 159.·49-s + 44.3·53-s − 989.·55-s + 324·59-s + 324·61-s + 131.·65-s − 464.·67-s + 1.05e3·71-s + 1.02e3·73-s + 1.20e3·77-s + 602.·79-s + 15.8·83-s + ⋯ |
L(s) = 1 | + 1.64·5-s − 1.21·7-s − 1.47·11-s + 0.152·13-s − 0.566·17-s + 1.51·19-s + 0.898·23-s + 1.71·25-s − 1.31·29-s − 0.854·31-s − 1.99·35-s + 0.558·37-s + 1.92·41-s − 1.46·43-s − 0.974·47-s + 0.465·49-s + 0.114·53-s − 2.42·55-s + 0.714·59-s + 0.680·61-s + 0.251·65-s − 0.846·67-s + 1.75·71-s + 1.63·73-s + 1.78·77-s + 0.858·79-s + 0.0209·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.263186450\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.263186450\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 - 18.4T + 125T^{2} \) |
| 7 | \( 1 + 22.4T + 343T^{2} \) |
| 11 | \( 1 + 53.6T + 1.33e3T^{2} \) |
| 13 | \( 1 - 7.15T + 2.19e3T^{2} \) |
| 17 | \( 1 + 39.6T + 4.91e3T^{2} \) |
| 19 | \( 1 - 125.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 99.1T + 1.21e4T^{2} \) |
| 29 | \( 1 + 205.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 147.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 125.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 506.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 413.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 313.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 44.3T + 1.48e5T^{2} \) |
| 59 | \( 1 - 324T + 2.05e5T^{2} \) |
| 61 | \( 1 - 324T + 2.26e5T^{2} \) |
| 67 | \( 1 + 464.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.05e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.02e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 602.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 15.8T + 5.71e5T^{2} \) |
| 89 | \( 1 + 381.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 659.T + 9.12e5T^{2} \) |
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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
−8.943878031168676628002083567202, −7.79578008172902258113184319728, −6.99119634526765298131938463513, −6.23815458974068237912577758175, −5.48945793398630646964455132988, −5.05850200124329891969899282229, −3.49807477799745060304493014767, −2.74636089302275278230753573664, −1.96294960659132840244103079622, −0.65092672914262394498831358499,
0.65092672914262394498831358499, 1.96294960659132840244103079622, 2.74636089302275278230753573664, 3.49807477799745060304493014767, 5.05850200124329891969899282229, 5.48945793398630646964455132988, 6.23815458974068237912577758175, 6.99119634526765298131938463513, 7.79578008172902258113184319728, 8.943878031168676628002083567202