| L(s) = 1 | + 0.255·2-s − 1.93·4-s − 2.57·5-s − 0.612·7-s − 1.00·8-s − 0.659·10-s + 11-s − 3.42·13-s − 0.156·14-s + 3.61·16-s − 1.40·17-s − 0.203·19-s + 4.98·20-s + 0.255·22-s + 7.26·23-s + 1.64·25-s − 0.874·26-s + 1.18·28-s + 2.39·29-s + 0.0531·31-s + 2.93·32-s − 0.360·34-s + 1.57·35-s − 4.56·37-s − 0.0520·38-s + 2.59·40-s + 12.5·41-s + ⋯ |
| L(s) = 1 | + 0.180·2-s − 0.967·4-s − 1.15·5-s − 0.231·7-s − 0.355·8-s − 0.208·10-s + 0.301·11-s − 0.948·13-s − 0.0418·14-s + 0.902·16-s − 0.341·17-s − 0.0467·19-s + 1.11·20-s + 0.0545·22-s + 1.51·23-s + 0.329·25-s − 0.171·26-s + 0.223·28-s + 0.444·29-s + 0.00955·31-s + 0.518·32-s − 0.0617·34-s + 0.266·35-s − 0.751·37-s − 0.00844·38-s + 0.410·40-s + 1.96·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 - T \) |
| 61 | \( 1 + T \) |
| good | 2 | \( 1 - 0.255T + 2T^{2} \) |
| 5 | \( 1 + 2.57T + 5T^{2} \) |
| 7 | \( 1 + 0.612T + 7T^{2} \) |
| 13 | \( 1 + 3.42T + 13T^{2} \) |
| 17 | \( 1 + 1.40T + 17T^{2} \) |
| 19 | \( 1 + 0.203T + 19T^{2} \) |
| 23 | \( 1 - 7.26T + 23T^{2} \) |
| 29 | \( 1 - 2.39T + 29T^{2} \) |
| 31 | \( 1 - 0.0531T + 31T^{2} \) |
| 37 | \( 1 + 4.56T + 37T^{2} \) |
| 41 | \( 1 - 12.5T + 41T^{2} \) |
| 43 | \( 1 - 5.84T + 43T^{2} \) |
| 47 | \( 1 + 5.88T + 47T^{2} \) |
| 53 | \( 1 - 9.59T + 53T^{2} \) |
| 59 | \( 1 - 6.40T + 59T^{2} \) |
| 67 | \( 1 + 6.19T + 67T^{2} \) |
| 71 | \( 1 + 6.32T + 71T^{2} \) |
| 73 | \( 1 + 3.28T + 73T^{2} \) |
| 79 | \( 1 - 6.61T + 79T^{2} \) |
| 83 | \( 1 - 6.24T + 83T^{2} \) |
| 89 | \( 1 + 1.31T + 89T^{2} \) |
| 97 | \( 1 - 6.95T + 97T^{2} \) |
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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
−7.67003396620578863150134060051, −7.19713385344660066394185513417, −6.32053141461632382676139496692, −5.35526756858032512924829904509, −4.69152078492557978290871778006, −4.13181115757328668660619166328, −3.39896632862476179554682207728, −2.58010308991557675262260552374, −0.989592340314482323770191671701, 0,
0.989592340314482323770191671701, 2.58010308991557675262260552374, 3.39896632862476179554682207728, 4.13181115757328668660619166328, 4.69152078492557978290871778006, 5.35526756858032512924829904509, 6.32053141461632382676139496692, 7.19713385344660066394185513417, 7.67003396620578863150134060051
