L(s) = 1 | + 2-s + 4-s − 3·5-s + 8-s − 3·10-s + 11-s + 6·13-s + 16-s + 5·17-s + 6·19-s − 3·20-s + 22-s − 5·23-s + 4·25-s + 6·26-s + 6·29-s + 4·31-s + 32-s + 5·34-s − 2·37-s + 6·38-s − 3·40-s − 5·41-s − 10·43-s + 44-s − 5·46-s − 9·47-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s − 1.34·5-s + 0.353·8-s − 0.948·10-s + 0.301·11-s + 1.66·13-s + 1/4·16-s + 1.21·17-s + 1.37·19-s − 0.670·20-s + 0.213·22-s − 1.04·23-s + 4/5·25-s + 1.17·26-s + 1.11·29-s + 0.718·31-s + 0.176·32-s + 0.857·34-s − 0.328·37-s + 0.973·38-s − 0.474·40-s − 0.780·41-s − 1.52·43-s + 0.150·44-s − 0.737·46-s − 1.31·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.110004611\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.110004611\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
good | 5 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 12 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 9 T + p T^{2} \) |
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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
−7.78985164827121356958698172807, −6.86969579364814092774886375109, −6.36432859325799059918110336209, −5.54423582813924940476279682580, −4.86993200973878544674445894406, −4.03164938769803071734816121409, −3.44808584822434878005567436131, −3.13351431247508337237661039953, −1.63656291905460921682623916998, −0.803813792845549000530706136404,
0.803813792845549000530706136404, 1.63656291905460921682623916998, 3.13351431247508337237661039953, 3.44808584822434878005567436131, 4.03164938769803071734816121409, 4.86993200973878544674445894406, 5.54423582813924940476279682580, 6.36432859325799059918110336209, 6.86969579364814092774886375109, 7.78985164827121356958698172807