L(s) = 1 | − 3-s − 5-s − 2·9-s − 6·11-s − 4·13-s + 15-s − 2·19-s + 3·23-s + 25-s + 5·27-s + 3·29-s + 8·31-s + 6·33-s + 4·37-s + 4·39-s − 9·41-s − 7·43-s + 2·45-s + 6·53-s + 6·55-s + 2·57-s + 6·59-s + 5·61-s + 4·65-s + 5·67-s − 3·69-s + 6·71-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s − 2/3·9-s − 1.80·11-s − 1.10·13-s + 0.258·15-s − 0.458·19-s + 0.625·23-s + 1/5·25-s + 0.962·27-s + 0.557·29-s + 1.43·31-s + 1.04·33-s + 0.657·37-s + 0.640·39-s − 1.40·41-s − 1.06·43-s + 0.298·45-s + 0.824·53-s + 0.809·55-s + 0.264·57-s + 0.781·59-s + 0.640·61-s + 0.496·65-s + 0.610·67-s − 0.361·69-s + 0.712·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 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
−16.28095127247319, −15.74385747239284, −15.10829335126620, −14.86330306517881, −13.99928301339455, −13.41957455731746, −12.90425772712353, −12.23142759998565, −11.86052666608810, −11.20836920337390, −10.66692669750337, −10.13123501709273, −9.669833457571787, −8.545489765634843, −8.286905486974323, −7.685932782399964, −6.888302390408729, −6.428608900179182, −5.443643437919346, −5.068223308113141, −4.627576720400164, −3.517338518318765, −2.701900821096220, −2.320129580860657, −0.7784328724766832, 0,
0.7784328724766832, 2.320129580860657, 2.701900821096220, 3.517338518318765, 4.627576720400164, 5.068223308113141, 5.443643437919346, 6.428608900179182, 6.888302390408729, 7.685932782399964, 8.286905486974323, 8.545489765634843, 9.669833457571787, 10.13123501709273, 10.66692669750337, 11.20836920337390, 11.86052666608810, 12.23142759998565, 12.90425772712353, 13.41957455731746, 13.99928301339455, 14.86330306517881, 15.10829335126620, 15.74385747239284, 16.28095127247319