L(s) = 1 | + 1.76·3-s − 2.22·5-s + 2.22·7-s + 0.111·9-s + 11-s + 1.43·13-s − 3.91·15-s − 7.55·17-s − 4.84·19-s + 3.93·21-s + 7.89·23-s − 0.0613·25-s − 5.09·27-s + 3.77·29-s + 10.4·31-s + 1.76·33-s − 4.95·35-s + 7.82·37-s + 2.52·39-s + 6.59·41-s − 5.22·43-s − 0.247·45-s + 1.50·47-s − 2.03·49-s − 13.3·51-s − 5.32·53-s − 2.22·55-s + ⋯ |
L(s) = 1 | + 1.01·3-s − 0.993·5-s + 0.842·7-s + 0.0371·9-s + 0.301·11-s + 0.397·13-s − 1.01·15-s − 1.83·17-s − 1.11·19-s + 0.857·21-s + 1.64·23-s − 0.0122·25-s − 0.980·27-s + 0.700·29-s + 1.87·31-s + 0.307·33-s − 0.837·35-s + 1.28·37-s + 0.404·39-s + 1.02·41-s − 0.796·43-s − 0.0368·45-s + 0.220·47-s − 0.290·49-s − 1.86·51-s − 0.731·53-s − 0.299·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6028 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.433828638\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.433828638\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 11 | \( 1 - T \) |
| 137 | \( 1 - T \) |
good | 3 | \( 1 - 1.76T + 3T^{2} \) |
| 5 | \( 1 + 2.22T + 5T^{2} \) |
| 7 | \( 1 - 2.22T + 7T^{2} \) |
| 13 | \( 1 - 1.43T + 13T^{2} \) |
| 17 | \( 1 + 7.55T + 17T^{2} \) |
| 19 | \( 1 + 4.84T + 19T^{2} \) |
| 23 | \( 1 - 7.89T + 23T^{2} \) |
| 29 | \( 1 - 3.77T + 29T^{2} \) |
| 31 | \( 1 - 10.4T + 31T^{2} \) |
| 37 | \( 1 - 7.82T + 37T^{2} \) |
| 41 | \( 1 - 6.59T + 41T^{2} \) |
| 43 | \( 1 + 5.22T + 43T^{2} \) |
| 47 | \( 1 - 1.50T + 47T^{2} \) |
| 53 | \( 1 + 5.32T + 53T^{2} \) |
| 59 | \( 1 - 12.3T + 59T^{2} \) |
| 61 | \( 1 - 5.25T + 61T^{2} \) |
| 67 | \( 1 + 2.54T + 67T^{2} \) |
| 71 | \( 1 - 4.84T + 71T^{2} \) |
| 73 | \( 1 + 10.3T + 73T^{2} \) |
| 79 | \( 1 - 0.949T + 79T^{2} \) |
| 83 | \( 1 - 10.6T + 83T^{2} \) |
| 89 | \( 1 - 14.4T + 89T^{2} \) |
| 97 | \( 1 - 10.0T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.181653980070480904632838970735, −7.62879442568076353049070095313, −6.73567574694807569719519286791, −6.16842365741380400676787056209, −4.76238331859898703742244679305, −4.46089686703075705799799190973, −3.65668590719740351052967839034, −2.74214135730530905679995663946, −2.08318777675803370232219511993, −0.77434440161416036497136137327,
0.77434440161416036497136137327, 2.08318777675803370232219511993, 2.74214135730530905679995663946, 3.65668590719740351052967839034, 4.46089686703075705799799190973, 4.76238331859898703742244679305, 6.16842365741380400676787056209, 6.73567574694807569719519286791, 7.62879442568076353049070095313, 8.181653980070480904632838970735