L(s) = 1 | − 1.41·2-s − 2.64·5-s + 7-s + 2.82·8-s + 3.74·10-s + 1.74·13-s − 1.41·14-s − 4.00·16-s − 0.182·17-s − 1.74·19-s + 6.70·23-s + 2.00·25-s − 2.46·26-s + 6.70·29-s − 9.48·31-s + 0.258·34-s − 2.64·35-s − 11.4·37-s + 2.46·38-s − 7.48·40-s + 5.65·41-s − 43-s − 9.48·46-s − 0.182·47-s + 49-s + ⋯ |
L(s) = 1 | − 1.00·2-s − 1.18·5-s + 0.377·7-s + 0.999·8-s + 1.18·10-s + 0.483·13-s − 0.377·14-s − 1.00·16-s − 0.0443·17-s − 0.399·19-s + 1.39·23-s + 0.400·25-s − 0.483·26-s + 1.24·29-s − 1.70·31-s + 0.0443·34-s − 0.447·35-s − 1.88·37-s + 0.399·38-s − 1.18·40-s + 0.883·41-s − 0.152·43-s − 1.39·46-s − 0.0266·47-s + 0.142·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + 1.41T + 2T^{2} \) |
| 5 | \( 1 + 2.64T + 5T^{2} \) |
| 13 | \( 1 - 1.74T + 13T^{2} \) |
| 17 | \( 1 + 0.182T + 17T^{2} \) |
| 19 | \( 1 + 1.74T + 19T^{2} \) |
| 23 | \( 1 - 6.70T + 23T^{2} \) |
| 29 | \( 1 - 6.70T + 29T^{2} \) |
| 31 | \( 1 + 9.48T + 31T^{2} \) |
| 37 | \( 1 + 11.4T + 37T^{2} \) |
| 41 | \( 1 - 5.65T + 41T^{2} \) |
| 43 | \( 1 + T + 43T^{2} \) |
| 47 | \( 1 + 0.182T + 47T^{2} \) |
| 53 | \( 1 + 7.75T + 53T^{2} \) |
| 59 | \( 1 - 3.01T + 59T^{2} \) |
| 61 | \( 1 + 5.74T + 61T^{2} \) |
| 67 | \( 1 - 8.48T + 67T^{2} \) |
| 71 | \( 1 + 1.41T + 71T^{2} \) |
| 73 | \( 1 + 8.25T + 73T^{2} \) |
| 79 | \( 1 - 1.48T + 79T^{2} \) |
| 83 | \( 1 + 5.83T + 83T^{2} \) |
| 89 | \( 1 - 11.1T + 89T^{2} \) |
| 97 | \( 1 - 15.2T + 97T^{2} \) |
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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
−7.61851023198741808159046617958, −7.20757450928353023635262912806, −6.41223525360055875069568733354, −5.24291714489138113801193643407, −4.67794111145984956255797310069, −3.90800592944588166077255711569, −3.21948187128778142334188504910, −1.93086024471278529976048115838, −0.994143114623227906791068822615, 0,
0.994143114623227906791068822615, 1.93086024471278529976048115838, 3.21948187128778142334188504910, 3.90800592944588166077255711569, 4.67794111145984956255797310069, 5.24291714489138113801193643407, 6.41223525360055875069568733354, 7.20757450928353023635262912806, 7.61851023198741808159046617958