L(s) = 1 | − 2·3-s + 2·5-s + 7-s + 9-s − 6·11-s + 4·13-s − 4·15-s − 2·17-s − 4·19-s − 2·21-s + 23-s − 25-s + 4·27-s + 10·29-s − 8·31-s + 12·33-s + 2·35-s + 8·37-s − 8·39-s − 2·41-s − 6·43-s + 2·45-s + 12·47-s + 49-s + 4·51-s − 12·53-s − 12·55-s + ⋯ |
L(s) = 1 | − 1.15·3-s + 0.894·5-s + 0.377·7-s + 1/3·9-s − 1.80·11-s + 1.10·13-s − 1.03·15-s − 0.485·17-s − 0.917·19-s − 0.436·21-s + 0.208·23-s − 1/5·25-s + 0.769·27-s + 1.85·29-s − 1.43·31-s + 2.08·33-s + 0.338·35-s + 1.31·37-s − 1.28·39-s − 0.312·41-s − 0.914·43-s + 0.298·45-s + 1.75·47-s + 1/7·49-s + 0.560·51-s − 1.64·53-s − 1.61·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−17.11394405060462, −16.22555451697840, −15.93867934292886, −15.33993118304071, −14.58854008453860, −13.82958758377130, −13.40904527912601, −12.80858750541277, −12.38958666565260, −11.45761352750044, −10.90790669717707, −10.69317930655354, −10.06770770456393, −9.323795413567404, −8.344524364265233, −8.192634383427401, −7.070996025013363, −6.400852131753493, −5.902556694820503, −5.311810109989738, −4.857571092303671, −3.966950946936078, −2.769783455205004, −2.159168990959696, −1.079023234392304, 0,
1.079023234392304, 2.159168990959696, 2.769783455205004, 3.966950946936078, 4.857571092303671, 5.311810109989738, 5.902556694820503, 6.400852131753493, 7.070996025013363, 8.192634383427401, 8.344524364265233, 9.323795413567404, 10.06770770456393, 10.69317930655354, 10.90790669717707, 11.45761352750044, 12.38958666565260, 12.80858750541277, 13.40904527912601, 13.82958758377130, 14.58854008453860, 15.33993118304071, 15.93867934292886, 16.22555451697840, 17.11394405060462