L(s) = 1 | − i·2-s − 4-s + i·8-s + (3 + 1.41i)11-s + 4.24·13-s + 16-s + 6i·17-s + 4.24i·19-s + (1.41 − 3i)22-s + 1.41·23-s − 4.24i·26-s + 2·31-s − i·32-s + 6·34-s + 10i·37-s + 4.24·38-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 0.5·4-s + 0.353i·8-s + (0.904 + 0.426i)11-s + 1.17·13-s + 0.250·16-s + 1.45i·17-s + 0.973i·19-s + (0.301 − 0.639i)22-s + 0.294·23-s − 0.832i·26-s + 0.359·31-s − 0.176i·32-s + 1.02·34-s + 1.64i·37-s + 0.688·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.558 - 0.829i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.558 - 0.829i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.516487154\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.516487154\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 + (-3 - 1.41i)T \) |
good | 7 | \( 1 + 7T^{2} \) |
| 13 | \( 1 - 4.24T + 13T^{2} \) |
| 17 | \( 1 - 6iT - 17T^{2} \) |
| 19 | \( 1 - 4.24iT - 19T^{2} \) |
| 23 | \( 1 - 1.41T + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 - 10iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 12.7T + 43T^{2} \) |
| 47 | \( 1 + 9.89T + 47T^{2} \) |
| 53 | \( 1 - 5.65T + 53T^{2} \) |
| 59 | \( 1 - 5.65iT - 59T^{2} \) |
| 61 | \( 1 + 12.7iT - 61T^{2} \) |
| 67 | \( 1 + 8iT - 67T^{2} \) |
| 71 | \( 1 - 15.5iT - 71T^{2} \) |
| 73 | \( 1 + 8.48T + 73T^{2} \) |
| 79 | \( 1 + 8.48iT - 79T^{2} \) |
| 83 | \( 1 - 6iT - 83T^{2} \) |
| 89 | \( 1 + 7.07iT - 89T^{2} \) |
| 97 | \( 1 - 4iT - 97T^{2} \) |
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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
−8.332608506708861651875852837368, −8.043606477134857213511911201307, −6.58839466782060077622052848448, −6.40889999731721546486793833312, −5.34719340481156292167152559755, −4.48158929592092706894981900178, −3.68818600268904710914117561145, −3.22414737266088675312178555354, −1.71712754054003001545701124836, −1.39104364352773461536185236979,
0.42667616366339673582518813603, 1.52777766056305040475487363757, 2.94980975689300899093636849435, 3.65604252222718450744563414059, 4.58350734700351960257523931998, 5.26318878916002941604994476943, 6.08295801404304468478209406345, 6.78901348843369435871406841068, 7.18938135096483835077821698108, 8.261615950640564144759706812211