L(s) = 1 | − 5-s + 7-s − 4·11-s + 6·13-s − 4·17-s − 6·19-s + 25-s + 6·29-s + 4·31-s − 35-s + 8·37-s − 10·41-s + 2·43-s + 10·47-s + 49-s − 14·53-s + 4·55-s − 4·59-s − 8·61-s − 6·65-s − 6·67-s − 2·71-s − 10·73-s − 4·77-s − 16·79-s − 8·83-s + 4·85-s + ⋯ |
L(s) = 1 | − 0.447·5-s + 0.377·7-s − 1.20·11-s + 1.66·13-s − 0.970·17-s − 1.37·19-s + 1/5·25-s + 1.11·29-s + 0.718·31-s − 0.169·35-s + 1.31·37-s − 1.56·41-s + 0.304·43-s + 1.45·47-s + 1/7·49-s − 1.92·53-s + 0.539·55-s − 0.520·59-s − 1.02·61-s − 0.744·65-s − 0.733·67-s − 0.237·71-s − 1.17·73-s − 0.455·77-s − 1.80·79-s − 0.878·83-s + 0.433·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 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
−8.069611383414772741774693302770, −7.21799695752445847234181122208, −6.30416321132775573309989296436, −5.89696995572428970847916206455, −4.62126566439836026484724539423, −4.39969790073893561134465001213, −3.25403913705884389617061970832, −2.45297483335503941614766467417, −1.34316536281700637721265528471, 0,
1.34316536281700637721265528471, 2.45297483335503941614766467417, 3.25403913705884389617061970832, 4.39969790073893561134465001213, 4.62126566439836026484724539423, 5.89696995572428970847916206455, 6.30416321132775573309989296436, 7.21799695752445847234181122208, 8.069611383414772741774693302770