| L(s) = 1 | − 2-s − 4-s − 5-s − 3·7-s + 3·8-s + 10-s − 11-s + 6·13-s + 3·14-s − 16-s + 20-s + 22-s + 2·23-s − 4·25-s − 6·26-s + 3·28-s + 4·29-s − 4·31-s − 5·32-s + 3·35-s − 7·37-s − 3·40-s − 4·41-s − 4·43-s + 44-s − 2·46-s − 6·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s − 0.447·5-s − 1.13·7-s + 1.06·8-s + 0.316·10-s − 0.301·11-s + 1.66·13-s + 0.801·14-s − 1/4·16-s + 0.223·20-s + 0.213·22-s + 0.417·23-s − 4/5·25-s − 1.17·26-s + 0.566·28-s + 0.742·29-s − 0.718·31-s − 0.883·32-s + 0.507·35-s − 1.15·37-s − 0.474·40-s − 0.624·41-s − 0.609·43-s + 0.150·44-s − 0.294·46-s − 0.875·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35739 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35739 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6124131621\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6124131621\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 15 T + p T^{2} \) |
| 83 | \( 1 - 17 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−15.15113928991086, −14.24157761700226, −13.71750273107825, −13.44765274858832, −12.78779728419490, −12.50759130648335, −11.58439649415785, −11.16806892550125, −10.51844815868013, −10.10673897258242, −9.548402128944840, −9.025897893470694, −8.422747183571228, −8.177127360312997, −7.426513605978292, −6.698181966306976, −6.386596599482596, −5.445658645057304, −5.036428008718672, −3.976185739331690, −3.706871006649308, −3.120397503346888, −2.022081850747118, −1.190420933121918, −0.3720706392916795,
0.3720706392916795, 1.190420933121918, 2.022081850747118, 3.120397503346888, 3.706871006649308, 3.976185739331690, 5.036428008718672, 5.445658645057304, 6.386596599482596, 6.698181966306976, 7.426513605978292, 8.177127360312997, 8.422747183571228, 9.025897893470694, 9.548402128944840, 10.10673897258242, 10.51844815868013, 11.16806892550125, 11.58439649415785, 12.50759130648335, 12.78779728419490, 13.44765274858832, 13.71750273107825, 14.24157761700226, 15.15113928991086
