| L(s) = 1 | + 2-s − 4-s − 4·7-s − 3·8-s − 4·11-s + 2·13-s − 4·14-s − 16-s + 4·19-s − 4·22-s + 4·23-s − 5·25-s + 2·26-s + 4·28-s + 4·31-s + 5·32-s − 8·37-s + 4·38-s + 8·41-s + 4·43-s + 4·44-s + 4·46-s + 8·47-s + 9·49-s − 5·50-s − 2·52-s + 6·53-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 1.51·7-s − 1.06·8-s − 1.20·11-s + 0.554·13-s − 1.06·14-s − 1/4·16-s + 0.917·19-s − 0.852·22-s + 0.834·23-s − 25-s + 0.392·26-s + 0.755·28-s + 0.718·31-s + 0.883·32-s − 1.31·37-s + 0.648·38-s + 1.24·41-s + 0.609·43-s + 0.603·44-s + 0.589·46-s + 1.16·47-s + 9/7·49-s − 0.707·50-s − 0.277·52-s + 0.824·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.286908191\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.286908191\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−9.080812825182643628500386241988, −8.101143442390922056379274957023, −7.27367363457446830165141100633, −6.32405977600746485733429410262, −5.70745184069091360527312682431, −5.03751036887489134846457093232, −3.97199114657563656658945686348, −3.27721619631487510651281686768, −2.58741197616451373067915674525, −0.61845716452994561861133979136,
0.61845716452994561861133979136, 2.58741197616451373067915674525, 3.27721619631487510651281686768, 3.97199114657563656658945686348, 5.03751036887489134846457093232, 5.70745184069091360527312682431, 6.32405977600746485733429410262, 7.27367363457446830165141100633, 8.101143442390922056379274957023, 9.080812825182643628500386241988
