L(s) = 1 | − 2.44·2-s + 1.73·3-s + 3.99·4-s − 4.24·6-s − 4.89·8-s + 5·11-s + 6.92·12-s − 1.73·13-s + 3.99·16-s − 1.73·17-s + 2.82·19-s − 12.2·22-s + 2.44·23-s − 8.48·24-s + 4.24·26-s − 5.19·27-s + 5·29-s + 1.41·31-s + 8.66·33-s + 4.24·34-s − 2.44·37-s − 6.92·38-s − 2.99·39-s + 9.89·41-s + 19.9·44-s + ⋯ |
L(s) = 1 | − 1.73·2-s + 1.00·3-s + 1.99·4-s − 1.73·6-s − 1.73·8-s + 1.50·11-s + 1.99·12-s − 0.480·13-s + 0.999·16-s − 0.420·17-s + 0.648·19-s − 2.61·22-s + 0.510·23-s − 1.73·24-s + 0.832·26-s − 1.00·27-s + 0.928·29-s + 0.254·31-s + 1.50·33-s + 0.727·34-s − 0.402·37-s − 1.12·38-s − 0.480·39-s + 1.54·41-s + 3.01·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.076529983\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.076529983\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + 2.44T + 2T^{2} \) |
| 3 | \( 1 - 1.73T + 3T^{2} \) |
| 11 | \( 1 - 5T + 11T^{2} \) |
| 13 | \( 1 + 1.73T + 13T^{2} \) |
| 17 | \( 1 + 1.73T + 17T^{2} \) |
| 19 | \( 1 - 2.82T + 19T^{2} \) |
| 23 | \( 1 - 2.44T + 23T^{2} \) |
| 29 | \( 1 - 5T + 29T^{2} \) |
| 31 | \( 1 - 1.41T + 31T^{2} \) |
| 37 | \( 1 + 2.44T + 37T^{2} \) |
| 41 | \( 1 - 9.89T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 - 8.66T + 47T^{2} \) |
| 53 | \( 1 - 7.34T + 53T^{2} \) |
| 59 | \( 1 + 1.41T + 59T^{2} \) |
| 61 | \( 1 + 2.82T + 61T^{2} \) |
| 67 | \( 1 + 7.34T + 67T^{2} \) |
| 71 | \( 1 - 2T + 71T^{2} \) |
| 73 | \( 1 - 13.8T + 73T^{2} \) |
| 79 | \( 1 + 3T + 79T^{2} \) |
| 83 | \( 1 + 13.8T + 83T^{2} \) |
| 89 | \( 1 + 16.9T + 89T^{2} \) |
| 97 | \( 1 - 1.73T + 97T^{2} \) |
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
−9.420581606493617854857620512716, −8.945405914805871354581152504390, −8.412175281916777198032561576725, −7.47530707875109561928766938252, −6.92412751221013539088507898746, −5.92228221993652917812740790214, −4.31233993018665969645733287494, −3.08501245842654506023146409902, −2.17235354459433419939620217231, −0.986820630442778391752820267778,
0.986820630442778391752820267778, 2.17235354459433419939620217231, 3.08501245842654506023146409902, 4.31233993018665969645733287494, 5.92228221993652917812740790214, 6.92412751221013539088507898746, 7.47530707875109561928766938252, 8.412175281916777198032561576725, 8.945405914805871354581152504390, 9.420581606493617854857620512716