L(s) = 1 | − 2·3-s + 5-s − 7-s + 9-s − 3·11-s − 2·15-s + 3·17-s + 6·19-s + 2·21-s − 2·23-s + 25-s + 4·27-s − 3·29-s − 3·31-s + 6·33-s − 35-s − 37-s + 3·41-s + 43-s + 45-s − 4·47-s − 6·49-s − 6·51-s + 13·53-s − 3·55-s − 12·57-s − 15·61-s + ⋯ |
L(s) = 1 | − 1.15·3-s + 0.447·5-s − 0.377·7-s + 1/3·9-s − 0.904·11-s − 0.516·15-s + 0.727·17-s + 1.37·19-s + 0.436·21-s − 0.417·23-s + 1/5·25-s + 0.769·27-s − 0.557·29-s − 0.538·31-s + 1.04·33-s − 0.169·35-s − 0.164·37-s + 0.468·41-s + 0.152·43-s + 0.149·45-s − 0.583·47-s − 6/7·49-s − 0.840·51-s + 1.78·53-s − 0.404·55-s − 1.58·57-s − 1.92·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 37 | \( 1 + T \) |
good | 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 13 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 15 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−8.270271306100592111239412072234, −7.49831009051549166645447960646, −6.76654201641600851167649939545, −5.77162846845092713605544691251, −5.55177590831197844606272911275, −4.75848365306684889407447061330, −3.52332963371328713101003529737, −2.62513400004584128215759559012, −1.26592217580020237546694300482, 0,
1.26592217580020237546694300482, 2.62513400004584128215759559012, 3.52332963371328713101003529737, 4.75848365306684889407447061330, 5.55177590831197844606272911275, 5.77162846845092713605544691251, 6.76654201641600851167649939545, 7.49831009051549166645447960646, 8.270271306100592111239412072234