L(s) = 1 | + 4·7-s − 11-s + 4·13-s − 2·17-s + 6·19-s − 6·23-s − 6·29-s − 10·37-s − 6·41-s − 10·43-s − 6·47-s + 9·49-s + 10·53-s + 12·67-s + 14·71-s + 2·73-s − 4·77-s − 8·79-s − 4·83-s + 16·91-s + 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s − 8·119-s + ⋯ |
L(s) = 1 | + 1.51·7-s − 0.301·11-s + 1.10·13-s − 0.485·17-s + 1.37·19-s − 1.25·23-s − 1.11·29-s − 1.64·37-s − 0.937·41-s − 1.52·43-s − 0.875·47-s + 9/7·49-s + 1.37·53-s + 1.46·67-s + 1.66·71-s + 0.234·73-s − 0.455·77-s − 0.900·79-s − 0.439·83-s + 1.67·91-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s − 0.733·119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 39600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 7 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 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
−15.09355728211065, −14.45954410803987, −13.98446061638131, −13.59708285966856, −13.15923159286895, −12.33659932295883, −11.75871686155705, −11.41313038217406, −11.04051230605642, −10.28858746967073, −9.932323582596211, −9.103655482005649, −8.528168851643794, −8.167555338141831, −7.679570859361630, −6.982356785509889, −6.439016983171873, −5.459533351747665, −5.332544234702101, −4.660483953667243, −3.722933330882754, −3.536018566617520, −2.345977420953926, −1.736796194753331, −1.206219862526191, 0,
1.206219862526191, 1.736796194753331, 2.345977420953926, 3.536018566617520, 3.722933330882754, 4.660483953667243, 5.332544234702101, 5.459533351747665, 6.439016983171873, 6.982356785509889, 7.679570859361630, 8.167555338141831, 8.528168851643794, 9.103655482005649, 9.932323582596211, 10.28858746967073, 11.04051230605642, 11.41313038217406, 11.75871686155705, 12.33659932295883, 13.15923159286895, 13.59708285966856, 13.98446061638131, 14.45954410803987, 15.09355728211065