L(s) = 1 | − 2-s + 4-s − 8-s − 2·11-s − 6·13-s + 16-s + 4·17-s − 8·19-s + 2·22-s + 6·26-s + 6·29-s + 8·31-s − 32-s − 4·34-s + 2·37-s + 8·38-s − 10·41-s − 4·43-s − 2·44-s − 8·47-s − 7·49-s − 6·52-s + 4·53-s − 6·58-s + 12·59-s − 10·61-s − 8·62-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s − 0.603·11-s − 1.66·13-s + 1/4·16-s + 0.970·17-s − 1.83·19-s + 0.426·22-s + 1.17·26-s + 1.11·29-s + 1.43·31-s − 0.176·32-s − 0.685·34-s + 0.328·37-s + 1.29·38-s − 1.56·41-s − 0.609·43-s − 0.301·44-s − 1.16·47-s − 49-s − 0.832·52-s + 0.549·53-s − 0.787·58-s + 1.56·59-s − 1.28·61-s − 1.01·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 167 | \( 1 + T \) |
good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−14.40486140708599, −13.92546822806996, −13.19193429710113, −12.66046356362962, −12.36222444792563, −11.72168510745536, −11.40335838269235, −10.48430513004232, −10.28969367470897, −9.832446511423950, −9.446919176974855, −8.518169751632115, −8.111631734309948, −8.021604881587667, −7.014935374449216, −6.738804467257493, −6.228550340866989, −5.340250275874235, −4.934953932108035, −4.402081965600890, −3.508136418935859, −2.816688791556347, −2.349072318783601, −1.715119822766701, −0.7229536094002055, 0,
0.7229536094002055, 1.715119822766701, 2.349072318783601, 2.816688791556347, 3.508136418935859, 4.402081965600890, 4.934953932108035, 5.340250275874235, 6.228550340866989, 6.738804467257493, 7.014935374449216, 8.021604881587667, 8.111631734309948, 8.518169751632115, 9.446919176974855, 9.832446511423950, 10.28969367470897, 10.48430513004232, 11.40335838269235, 11.72168510745536, 12.36222444792563, 12.66046356362962, 13.19193429710113, 13.92546822806996, 14.40486140708599