| L(s) = 1 | + i·2-s − 4-s + 5-s − i·8-s + i·10-s − 2.94i·11-s − 3.93i·13-s + 16-s − 0.399·17-s − 0.0352i·19-s − 20-s + 2.94·22-s + 3.73i·23-s + 25-s + 3.93·26-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s + 0.447·5-s − 0.353i·8-s + 0.316i·10-s − 0.888i·11-s − 1.09i·13-s + 0.250·16-s − 0.0969·17-s − 0.00809i·19-s − 0.223·20-s + 0.628·22-s + 0.778i·23-s + 0.200·25-s + 0.771·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.239 + 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.239 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7558297561\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7558297561\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 2.94iT - 11T^{2} \) |
| 13 | \( 1 + 3.93iT - 13T^{2} \) |
| 17 | \( 1 + 0.399T + 17T^{2} \) |
| 19 | \( 1 + 0.0352iT - 19T^{2} \) |
| 23 | \( 1 - 3.73iT - 23T^{2} \) |
| 29 | \( 1 + 8.89iT - 29T^{2} \) |
| 31 | \( 1 - 0.828iT - 31T^{2} \) |
| 37 | \( 1 + 7.93T + 37T^{2} \) |
| 41 | \( 1 + 6.31T + 41T^{2} \) |
| 43 | \( 1 + 3.03T + 43T^{2} \) |
| 47 | \( 1 + 5.80T + 47T^{2} \) |
| 53 | \( 1 - 4.29iT - 53T^{2} \) |
| 59 | \( 1 - 5.57T + 59T^{2} \) |
| 61 | \( 1 - 11.5iT - 61T^{2} \) |
| 67 | \( 1 + 12.5T + 67T^{2} \) |
| 71 | \( 1 + 1.93iT - 71T^{2} \) |
| 73 | \( 1 + 0.343iT - 73T^{2} \) |
| 79 | \( 1 - 8.30T + 79T^{2} \) |
| 83 | \( 1 + 10.3T + 83T^{2} \) |
| 89 | \( 1 + 6.17T + 89T^{2} \) |
| 97 | \( 1 + 15.6iT - 97T^{2} \) |
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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
−8.237511469363880406854184698367, −7.40465236472021014711326885008, −6.67833618381547469260617303723, −5.76733058535755856085858024602, −5.54877300411878842439202898722, −4.54730644383338347613845469265, −3.55328671938343936884557464544, −2.81011317504198323803751351811, −1.47491551381526961792532437827, −0.20323513376138647355049414048,
1.49275413420710860700703660923, 2.05913213471125807380785538604, 3.10144128010321635677955259948, 3.99061560353918855304619882179, 4.83669663958993716666896580234, 5.33421457916402227128479155486, 6.67812139101799869179469198627, 6.82740032593654124245879846380, 8.001287265135731976666020454820, 8.800250517354694543242626656440
