| L(s) = 1 | − i·3-s + i·7-s + 2·9-s − 5·11-s + i·13-s − 3i·17-s + 6·19-s + 21-s − 6i·23-s − 5i·27-s + 9·29-s + 5i·33-s − 6i·37-s + 39-s + 8·41-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + 0.377i·7-s + 0.666·9-s − 1.50·11-s + 0.277i·13-s − 0.727i·17-s + 1.37·19-s + 0.218·21-s − 1.25i·23-s − 0.962i·27-s + 1.67·29-s + 0.870i·33-s − 0.986i·37-s + 0.160·39-s + 1.24·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.603190838\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.603190838\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - iT \) |
| good | 3 | \( 1 + iT - 3T^{2} \) |
| 11 | \( 1 + 5T + 11T^{2} \) |
| 13 | \( 1 - iT - 13T^{2} \) |
| 17 | \( 1 + 3iT - 17T^{2} \) |
| 19 | \( 1 - 6T + 19T^{2} \) |
| 23 | \( 1 + 6iT - 23T^{2} \) |
| 29 | \( 1 - 9T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 6iT - 37T^{2} \) |
| 41 | \( 1 - 8T + 41T^{2} \) |
| 43 | \( 1 - 6iT - 43T^{2} \) |
| 47 | \( 1 + 3iT - 47T^{2} \) |
| 53 | \( 1 + 12iT - 53T^{2} \) |
| 59 | \( 1 + 8T + 59T^{2} \) |
| 61 | \( 1 + 4T + 61T^{2} \) |
| 67 | \( 1 - 4iT - 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 - 10iT - 73T^{2} \) |
| 79 | \( 1 - 3T + 79T^{2} \) |
| 83 | \( 1 + 12iT - 83T^{2} \) |
| 89 | \( 1 - 16T + 89T^{2} \) |
| 97 | \( 1 + 7iT - 97T^{2} \) |
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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
−9.511146756694063072849780364616, −8.454319508032720097249076904898, −7.74152208327946008940158465062, −7.09863865729534740582752256860, −6.21194920886477618236243061206, −5.19725513004886246870487322093, −4.50306578537665513108174397465, −3.01517848820079930900166197188, −2.24087465779208734826785086378, −0.75426739068348235660808495487,
1.21516411902723416399131658557, 2.77379127881215527383193628069, 3.66694085477657493600548533871, 4.73191917989119424612534923405, 5.33803774843133144785432928121, 6.38508060345397383102242753684, 7.64451692575950282623666912201, 7.77483863312673220349810341921, 9.074472203004943738961237653960, 9.830355631542343965210369418935
