L(s) = 1 | − i·2-s − i·3-s − 4-s − 6-s + i·7-s + i·8-s − 9-s + i·12-s − 2i·13-s + 14-s + 16-s − 6i·17-s + i·18-s − 8·19-s + 21-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 0.577i·3-s − 0.5·4-s − 0.408·6-s + 0.377i·7-s + 0.353i·8-s − 0.333·9-s + 0.288i·12-s − 0.554i·13-s + 0.267·14-s + 0.250·16-s − 1.45i·17-s + 0.235i·18-s − 1.83·19-s + 0.218·21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6393494667\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6393494667\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + iT \) |
| 5 | \( 1 \) |
| 7 | \( 1 - iT \) |
good | 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 + 6iT - 17T^{2} \) |
| 19 | \( 1 + 8T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + 10iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 - 10iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + 12iT - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 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
−9.359851012353017973039367266267, −8.780923119626631225513626118080, −7.84219075785560575448812641665, −7.00031212581663480313945020426, −5.92239313671592521064508708772, −5.09154103786102797841496572143, −3.93806844338426601468012027860, −2.77021536865459198252462738341, −1.90277964016715099523092025671, −0.27291155692953918357030306629,
1.90095491420799327919340864741, 3.63973708419947515253285945118, 4.24589462716121997227799806971, 5.24932878668138570319052657656, 6.26295393848384490139988056849, 6.86542985455919831086181389273, 8.078473581600251918966968228864, 8.597280079766162540289583543151, 9.473874051663781037138254672848, 10.37204067895360918481983228832