L(s) = 1 | + 2-s + 2·3-s − 4-s − 5-s + 2·6-s − 7-s − 3·8-s + 9-s − 10-s − 11-s − 2·12-s − 5·13-s − 14-s − 2·15-s − 16-s + 2·17-s + 18-s + 4·19-s + 20-s − 2·21-s − 22-s − 4·23-s − 6·24-s + 25-s − 5·26-s − 4·27-s + 28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.15·3-s − 1/2·4-s − 0.447·5-s + 0.816·6-s − 0.377·7-s − 1.06·8-s + 1/3·9-s − 0.316·10-s − 0.301·11-s − 0.577·12-s − 1.38·13-s − 0.267·14-s − 0.516·15-s − 1/4·16-s + 0.485·17-s + 0.235·18-s + 0.917·19-s + 0.223·20-s − 0.436·21-s − 0.213·22-s − 0.834·23-s − 1.22·24-s + 1/5·25-s − 0.980·26-s − 0.769·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.288692448\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.288692448\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 229 | \( 1 - T \) |
good | 2 | \( 1 - T + p T^{2} \) |
| 3 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 9 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 - T + p T^{2} \) |
| 43 | \( 1 + 3 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−7.898719321320502057648374756993, −7.36206953435314449863108224899, −6.37965963794771943990569544215, −5.56814222812180625871203977660, −4.90709785248513481855728210899, −4.17254320988831181485839243018, −3.47246234873361972124392891099, −2.89016808122029173224719965348, −2.24732134604970975343290490964, −0.60632871929251034792484372848,
0.60632871929251034792484372848, 2.24732134604970975343290490964, 2.89016808122029173224719965348, 3.47246234873361972124392891099, 4.17254320988831181485839243018, 4.90709785248513481855728210899, 5.56814222812180625871203977660, 6.37965963794771943990569544215, 7.36206953435314449863108224899, 7.898719321320502057648374756993