L(s) = 1 | − 3.16·3-s − 1.06·5-s − 3.52·7-s + 7.03·9-s − 2.93·11-s − 13-s + 3.38·15-s + 2.35·17-s − 2.71·19-s + 11.1·21-s + 1.38·23-s − 3.85·25-s − 12.7·27-s − 29-s + 3.37·31-s + 9.29·33-s + 3.77·35-s − 4.43·37-s + 3.16·39-s + 2.29·41-s + 1.23·43-s − 7.52·45-s + 4.44·47-s + 5.43·49-s − 7.47·51-s − 1.68·53-s + 3.13·55-s + ⋯ |
L(s) = 1 | − 1.82·3-s − 0.478·5-s − 1.33·7-s + 2.34·9-s − 0.884·11-s − 0.277·13-s + 0.874·15-s + 0.572·17-s − 0.621·19-s + 2.43·21-s + 0.288·23-s − 0.771·25-s − 2.46·27-s − 0.185·29-s + 0.605·31-s + 1.61·33-s + 0.637·35-s − 0.729·37-s + 0.507·39-s + 0.357·41-s + 0.187·43-s − 1.12·45-s + 0.647·47-s + 0.777·49-s − 1.04·51-s − 0.231·53-s + 0.422·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 13 | \( 1 + T \) |
| 29 | \( 1 + T \) |
good | 3 | \( 1 + 3.16T + 3T^{2} \) |
| 5 | \( 1 + 1.06T + 5T^{2} \) |
| 7 | \( 1 + 3.52T + 7T^{2} \) |
| 11 | \( 1 + 2.93T + 11T^{2} \) |
| 17 | \( 1 - 2.35T + 17T^{2} \) |
| 19 | \( 1 + 2.71T + 19T^{2} \) |
| 23 | \( 1 - 1.38T + 23T^{2} \) |
| 31 | \( 1 - 3.37T + 31T^{2} \) |
| 37 | \( 1 + 4.43T + 37T^{2} \) |
| 41 | \( 1 - 2.29T + 41T^{2} \) |
| 43 | \( 1 - 1.23T + 43T^{2} \) |
| 47 | \( 1 - 4.44T + 47T^{2} \) |
| 53 | \( 1 + 1.68T + 53T^{2} \) |
| 59 | \( 1 + 3.67T + 59T^{2} \) |
| 61 | \( 1 - 5.65T + 61T^{2} \) |
| 67 | \( 1 - 14.2T + 67T^{2} \) |
| 71 | \( 1 - 4.07T + 71T^{2} \) |
| 73 | \( 1 - 14.0T + 73T^{2} \) |
| 79 | \( 1 - 8.61T + 79T^{2} \) |
| 83 | \( 1 - 2.27T + 83T^{2} \) |
| 89 | \( 1 - 7.46T + 89T^{2} \) |
| 97 | \( 1 + 8.96T + 97T^{2} \) |
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
−7.51103468939934683520842397240, −6.78082258178002433543432710253, −6.32794914237416366253156926256, −5.56932917665603247773172724287, −5.06762999023887820823067478839, −4.15566844047701363752574849611, −3.43340658929270120414534046372, −2.26093130050488857402466515823, −0.78232403494764473402700346996, 0,
0.78232403494764473402700346996, 2.26093130050488857402466515823, 3.43340658929270120414534046372, 4.15566844047701363752574849611, 5.06762999023887820823067478839, 5.56932917665603247773172724287, 6.32794914237416366253156926256, 6.78082258178002433543432710253, 7.51103468939934683520842397240