L(s) = 1 | − 7-s + 4·11-s + 6·13-s + 2·17-s − 4·19-s − 8·23-s − 2·29-s − 10·37-s + 6·41-s + 4·43-s + 49-s − 6·53-s − 4·59-s − 6·61-s − 4·67-s + 8·71-s − 10·73-s − 4·77-s − 4·83-s + 6·89-s − 6·91-s + 14·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
L(s) = 1 | − 0.377·7-s + 1.20·11-s + 1.66·13-s + 0.485·17-s − 0.917·19-s − 1.66·23-s − 0.371·29-s − 1.64·37-s + 0.937·41-s + 0.609·43-s + 1/7·49-s − 0.824·53-s − 0.520·59-s − 0.768·61-s − 0.488·67-s + 0.949·71-s − 1.17·73-s − 0.455·77-s − 0.439·83-s + 0.635·89-s − 0.628·91-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−14.11425218958706, −13.51919560351908, −13.06750515727428, −12.39276847797258, −12.15256321364163, −11.56220301032159, −11.01397746521721, −10.60708417673627, −10.08301993488993, −9.503383762729043, −8.922305970399994, −8.687251871093289, −8.028912909488106, −7.521314267883096, −6.822004006587370, −6.197068451776775, −6.101995135298637, −5.498326715122532, −4.502370834034621, −4.122730200616865, −3.552866444439616, −3.197319987710577, −2.102924329634867, −1.651703041116161, −0.9445239170805478, 0,
0.9445239170805478, 1.651703041116161, 2.102924329634867, 3.197319987710577, 3.552866444439616, 4.122730200616865, 4.502370834034621, 5.498326715122532, 6.101995135298637, 6.197068451776775, 6.822004006587370, 7.521314267883096, 8.028912909488106, 8.687251871093289, 8.922305970399994, 9.503383762729043, 10.08301993488993, 10.60708417673627, 11.01397746521721, 11.56220301032159, 12.15256321364163, 12.39276847797258, 13.06750515727428, 13.51919560351908, 14.11425218958706