L(s) = 1 | − i·2-s − 4-s + (1 + 2i)5-s + i·8-s + (2 − i)10-s + 4·11-s − 2i·13-s + 16-s + 6i·17-s + 19-s + (−1 − 2i)20-s − 4i·22-s + 6i·23-s + (−3 + 4i)25-s − 2·26-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 0.5·4-s + (0.447 + 0.894i)5-s + 0.353i·8-s + (0.632 − 0.316i)10-s + 1.20·11-s − 0.554i·13-s + 0.250·16-s + 1.45i·17-s + 0.229·19-s + (−0.223 − 0.447i)20-s − 0.852i·22-s + 1.25i·23-s + (−0.600 + 0.800i)25-s − 0.392·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.733652938\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.733652938\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-1 - 2i)T \) |
| 19 | \( 1 - T \) |
good | 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 - 6iT - 17T^{2} \) |
| 23 | \( 1 - 6iT - 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 6T + 31T^{2} \) |
| 37 | \( 1 + 10iT - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 6iT - 43T^{2} \) |
| 47 | \( 1 + 6iT - 47T^{2} \) |
| 53 | \( 1 - 10iT - 53T^{2} \) |
| 59 | \( 1 - 2T + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 - 8iT - 67T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 - 16iT - 73T^{2} \) |
| 79 | \( 1 - 14T + 79T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 - 4T + 89T^{2} \) |
| 97 | \( 1 + 10iT - 97T^{2} \) |
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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
−9.478560357936067581335384702495, −8.890682772364043322594933188267, −7.79239889737717952551858614211, −7.03425856364710671627961810961, −6.01562802053806258529100842932, −5.47487895113361789651879457349, −3.92788901675125513669078175942, −3.56038146481319896683270141331, −2.30199979476279768089729394756, −1.36304288652075625360353109695,
0.71739516910332683208464858455, 2.00935667495296722648840420197, 3.53694432865273450749967203236, 4.59238737507768974909141457509, 5.10082158543761354992029439073, 6.18611663618006758365874689405, 6.75796566230088970954744596744, 7.65820989500424078068735344126, 8.619524224170295971456713097223, 9.228620766441018194027331559874