| L(s) = 1 | − 4·2-s + 16·4-s − 233·7-s − 64·8-s + 498·11-s − 809·13-s + 932·14-s + 256·16-s − 1.00e3·17-s − 1.70e3·19-s − 1.99e3·22-s + 1.55e3·23-s + 3.23e3·26-s − 3.72e3·28-s − 7.83e3·29-s + 977·31-s − 1.02e3·32-s + 4.00e3·34-s + 4.82e3·37-s + 6.82e3·38-s + 8.14e3·41-s − 1.94e4·43-s + 7.96e3·44-s − 6.21e3·46-s + 8.41e3·47-s + 3.74e4·49-s − 1.29e4·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 1.79·7-s − 0.353·8-s + 1.24·11-s − 1.32·13-s + 1.27·14-s + 1/4·16-s − 0.840·17-s − 1.08·19-s − 0.877·22-s + 0.612·23-s + 0.938·26-s − 0.898·28-s − 1.72·29-s + 0.182·31-s − 0.176·32-s + 0.594·34-s + 0.579·37-s + 0.766·38-s + 0.756·41-s − 1.60·43-s + 0.620·44-s − 0.433·46-s + 0.555·47-s + 2.23·49-s − 0.663·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.5707197697\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5707197697\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{2} T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 233 T + p^{5} T^{2} \) |
| 11 | \( 1 - 498 T + p^{5} T^{2} \) |
| 13 | \( 1 + 809 T + p^{5} T^{2} \) |
| 17 | \( 1 + 1002 T + p^{5} T^{2} \) |
| 19 | \( 1 + 1705 T + p^{5} T^{2} \) |
| 23 | \( 1 - 1554 T + p^{5} T^{2} \) |
| 29 | \( 1 + 270 p T + p^{5} T^{2} \) |
| 31 | \( 1 - 977 T + p^{5} T^{2} \) |
| 37 | \( 1 - 4822 T + p^{5} T^{2} \) |
| 41 | \( 1 - 8148 T + p^{5} T^{2} \) |
| 43 | \( 1 + 19469 T + p^{5} T^{2} \) |
| 47 | \( 1 - 8418 T + p^{5} T^{2} \) |
| 53 | \( 1 - 17664 T + p^{5} T^{2} \) |
| 59 | \( 1 + 35910 T + p^{5} T^{2} \) |
| 61 | \( 1 - 3527 T + p^{5} T^{2} \) |
| 67 | \( 1 + 57473 T + p^{5} T^{2} \) |
| 71 | \( 1 - 7548 T + p^{5} T^{2} \) |
| 73 | \( 1 - 646 T + p^{5} T^{2} \) |
| 79 | \( 1 + 22720 T + p^{5} T^{2} \) |
| 83 | \( 1 - 11574 T + p^{5} T^{2} \) |
| 89 | \( 1 - 78960 T + p^{5} T^{2} \) |
| 97 | \( 1 + 54593 T + p^{5} T^{2} \) |
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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
−10.02832136813374142906994808804, −9.360809570524561774418869487983, −8.847155959181019906082067900021, −7.35115573507703824295157780106, −6.71455035159919843286694373184, −5.95746626149412412110626846663, −4.31505204212519605496719496808, −3.18028981102353402934686531580, −2.05699397995915542396521867647, −0.40938195097136405045742298689,
0.40938195097136405045742298689, 2.05699397995915542396521867647, 3.18028981102353402934686531580, 4.31505204212519605496719496808, 5.95746626149412412110626846663, 6.71455035159919843286694373184, 7.35115573507703824295157780106, 8.847155959181019906082067900021, 9.360809570524561774418869487983, 10.02832136813374142906994808804
