L(s) = 1 | + (−0.707 − 0.707i)2-s + 1.00i·4-s + (−0.707 + 0.707i)7-s + (0.707 − 0.707i)8-s − 0.828i·11-s + (−0.585 − 0.585i)13-s + 1.00·14-s − 1.00·16-s + (4.41 + 4.41i)17-s − 7.07i·19-s + (−0.585 + 0.585i)22-s + (−0.828 + 0.828i)23-s + 0.828i·26-s + (−0.707 − 0.707i)28-s − 8.82·29-s + ⋯ |
L(s) = 1 | + (−0.499 − 0.499i)2-s + 0.500i·4-s + (−0.267 + 0.267i)7-s + (0.250 − 0.250i)8-s − 0.249i·11-s + (−0.162 − 0.162i)13-s + 0.267·14-s − 0.250·16-s + (1.07 + 1.07i)17-s − 1.62i·19-s + (−0.124 + 0.124i)22-s + (−0.172 + 0.172i)23-s + 0.162i·26-s + (−0.133 − 0.133i)28-s − 1.63·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0618 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0618 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.061958238\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.061958238\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 + 0.707i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (0.707 - 0.707i)T \) |
good | 11 | \( 1 + 0.828iT - 11T^{2} \) |
| 13 | \( 1 + (0.585 + 0.585i)T + 13iT^{2} \) |
| 17 | \( 1 + (-4.41 - 4.41i)T + 17iT^{2} \) |
| 19 | \( 1 + 7.07iT - 19T^{2} \) |
| 23 | \( 1 + (0.828 - 0.828i)T - 23iT^{2} \) |
| 29 | \( 1 + 8.82T + 29T^{2} \) |
| 31 | \( 1 + 3.17T + 31T^{2} \) |
| 37 | \( 1 + (-8.07 + 8.07i)T - 37iT^{2} \) |
| 41 | \( 1 - 0.828iT - 41T^{2} \) |
| 43 | \( 1 + (-3.58 - 3.58i)T + 43iT^{2} \) |
| 47 | \( 1 + (-3.58 - 3.58i)T + 47iT^{2} \) |
| 53 | \( 1 + (-0.828 + 0.828i)T - 53iT^{2} \) |
| 59 | \( 1 + 8T + 59T^{2} \) |
| 61 | \( 1 - 9.41T + 61T^{2} \) |
| 67 | \( 1 + (2.41 - 2.41i)T - 67iT^{2} \) |
| 71 | \( 1 + 3.75iT - 71T^{2} \) |
| 73 | \( 1 + (6 + 6i)T + 73iT^{2} \) |
| 79 | \( 1 + 9.31iT - 79T^{2} \) |
| 83 | \( 1 + (-1.65 + 1.65i)T - 83iT^{2} \) |
| 89 | \( 1 - 9.31T + 89T^{2} \) |
| 97 | \( 1 + (-0.242 + 0.242i)T - 97iT^{2} \) |
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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
−8.669827660139424891742561325244, −7.70413223610336726068481374536, −7.30873197142435606293218073849, −6.13306589818614387452077428757, −5.56815889093120248179875097669, −4.43640181956095379047313711904, −3.56550488057576299344553345313, −2.74990076614155504863510322201, −1.75725168677132275018266508608, −0.46611510454590898647007515716,
1.00004516500147651667628390211, 2.15088657271021988618501190948, 3.37351232724570179105632955338, 4.21574381892640133108339444670, 5.31623076142919796900468160979, 5.84604812713721242148296533872, 6.75845001373547099228332786623, 7.55732240145586446133302422576, 7.88637565434205669242246254598, 8.903981985556281871466977344809