L(s) = 1 | + 2·2-s + 2·4-s − 3·7-s − 2·11-s − 13-s − 6·14-s − 4·16-s − 2·17-s − 4·22-s − 6·23-s − 2·26-s − 6·28-s + 10·29-s + 3·31-s − 8·32-s − 4·34-s − 2·37-s − 8·41-s + 43-s − 4·44-s − 12·46-s − 2·47-s + 2·49-s − 2·52-s − 4·53-s + 20·58-s − 10·59-s + ⋯ |
L(s) = 1 | + 1.41·2-s + 4-s − 1.13·7-s − 0.603·11-s − 0.277·13-s − 1.60·14-s − 16-s − 0.485·17-s − 0.852·22-s − 1.25·23-s − 0.392·26-s − 1.13·28-s + 1.85·29-s + 0.538·31-s − 1.41·32-s − 0.685·34-s − 0.328·37-s − 1.24·41-s + 0.152·43-s − 0.603·44-s − 1.76·46-s − 0.291·47-s + 2/7·49-s − 0.277·52-s − 0.549·53-s + 2.62·58-s − 1.30·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.114625719\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.114625719\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 \) |
good | 2 | \( 1 - p T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 17 T + p T^{2} \) |
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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
−13.90610926264404, −13.36208525939968, −13.20988445668809, −12.46113689171048, −12.20599103287209, −11.81975438675018, −11.13865514248895, −10.51107201511329, −9.968633830123795, −9.704110686067549, −8.868355083223294, −8.413714479053771, −7.818065960084459, −6.928235001322288, −6.719119310018445, −6.084081774216017, −5.745170988448954, −4.972706144194703, −4.565338120992652, −4.020389186732739, −3.315603432259092, −2.876686863580175, −2.423619705878697, −1.527226441004306, −0.2507789513562879,
0.2507789513562879, 1.527226441004306, 2.423619705878697, 2.876686863580175, 3.315603432259092, 4.020389186732739, 4.565338120992652, 4.972706144194703, 5.745170988448954, 6.084081774216017, 6.719119310018445, 6.928235001322288, 7.818065960084459, 8.413714479053771, 8.868355083223294, 9.704110686067549, 9.968633830123795, 10.51107201511329, 11.13865514248895, 11.81975438675018, 12.20599103287209, 12.46113689171048, 13.20988445668809, 13.36208525939968, 13.90610926264404