L(s) = 1 | + 1.56·3-s − 5-s − 0.561·9-s + 1.56·11-s − 0.438·13-s − 1.56·15-s + 0.438·17-s − 7.12·19-s − 3.12·23-s + 25-s − 5.56·27-s + 6.68·29-s + 2.43·33-s + 6·37-s − 0.684·39-s − 5.12·41-s − 0.876·43-s + 0.561·45-s − 8.68·47-s + 0.684·51-s − 5.12·53-s − 1.56·55-s − 11.1·57-s − 4·59-s − 15.3·61-s + 0.438·65-s − 10.2·67-s + ⋯ |
L(s) = 1 | + 0.901·3-s − 0.447·5-s − 0.187·9-s + 0.470·11-s − 0.121·13-s − 0.403·15-s + 0.106·17-s − 1.63·19-s − 0.651·23-s + 0.200·25-s − 1.07·27-s + 1.24·29-s + 0.424·33-s + 0.986·37-s − 0.109·39-s − 0.800·41-s − 0.133·43-s + 0.0837·45-s − 1.26·47-s + 0.0958·51-s − 0.703·53-s − 0.210·55-s − 1.47·57-s − 0.520·59-s − 1.96·61-s + 0.0543·65-s − 1.25·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 1.56T + 3T^{2} \) |
| 11 | \( 1 - 1.56T + 11T^{2} \) |
| 13 | \( 1 + 0.438T + 13T^{2} \) |
| 17 | \( 1 - 0.438T + 17T^{2} \) |
| 19 | \( 1 + 7.12T + 19T^{2} \) |
| 23 | \( 1 + 3.12T + 23T^{2} \) |
| 29 | \( 1 - 6.68T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 6T + 37T^{2} \) |
| 41 | \( 1 + 5.12T + 41T^{2} \) |
| 43 | \( 1 + 0.876T + 43T^{2} \) |
| 47 | \( 1 + 8.68T + 47T^{2} \) |
| 53 | \( 1 + 5.12T + 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 + 15.3T + 61T^{2} \) |
| 67 | \( 1 + 10.2T + 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 - 12.2T + 73T^{2} \) |
| 79 | \( 1 - 2.43T + 79T^{2} \) |
| 83 | \( 1 - 4T + 83T^{2} \) |
| 89 | \( 1 - 1.12T + 89T^{2} \) |
| 97 | \( 1 + 5.80T + 97T^{2} \) |
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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
−8.122152840363497163874171911188, −7.66961860114679651126489267335, −6.54328572484194140102944955130, −6.13108024965993165777572245944, −4.85598705645500092368332686501, −4.19111071326445571508763274720, −3.35545520121437657410584336392, −2.59403383504048912179386416757, −1.62049698385777854730253003062, 0,
1.62049698385777854730253003062, 2.59403383504048912179386416757, 3.35545520121437657410584336392, 4.19111071326445571508763274720, 4.85598705645500092368332686501, 6.13108024965993165777572245944, 6.54328572484194140102944955130, 7.66961860114679651126489267335, 8.122152840363497163874171911188