L(s) = 1 | + 31.1·7-s − 62.3i·13-s − 56i·19-s + 125·25-s − 155.·31-s + 436. i·37-s − 520i·43-s + 629·49-s − 935. i·61-s − 880i·67-s + 1.19e3·73-s + 1.09e3·79-s − 1.94e3i·91-s + 1.33e3·97-s − 1.02e3·103-s + ⋯ |
L(s) = 1 | + 1.68·7-s − 1.33i·13-s − 0.676i·19-s + 25-s − 0.903·31-s + 1.93i·37-s − 1.84i·43-s + 1.83·49-s − 1.96i·61-s − 1.60i·67-s + 1.90·73-s + 1.55·79-s − 2.23i·91-s + 1.39·97-s − 0.984·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.376640124\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.376640124\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 - 125T^{2} \) |
| 7 | \( 1 - 31.1T + 343T^{2} \) |
| 11 | \( 1 - 1.33e3T^{2} \) |
| 13 | \( 1 + 62.3iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 4.91e3T^{2} \) |
| 19 | \( 1 + 56iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 1.21e4T^{2} \) |
| 29 | \( 1 - 2.43e4T^{2} \) |
| 31 | \( 1 + 155.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 436. iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 6.89e4T^{2} \) |
| 43 | \( 1 + 520iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 1.03e5T^{2} \) |
| 53 | \( 1 - 1.48e5T^{2} \) |
| 59 | \( 1 - 2.05e5T^{2} \) |
| 61 | \( 1 + 935. iT - 2.26e5T^{2} \) |
| 67 | \( 1 + 880iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.19e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.09e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 5.71e5T^{2} \) |
| 89 | \( 1 + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.33e3T + 9.12e5T^{2} \) |
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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
−10.49912455618165190277473469522, −9.219628665599115125501251980461, −8.280421395757425529257094247782, −7.76331001373454494378940423164, −6.65586898992746637698457271351, −5.27128099269900918543880838778, −4.85642264996169452708530555221, −3.41093966499436541116066998530, −2.07427991504094383689732004954, −0.794340645454267792128082113077,
1.27866218438550636600268532293, 2.24831316230899120951329007367, 3.95357947718611103681208369815, 4.76443069749800868349116880033, 5.72967424496039947151949618367, 6.96998242271999813662195366304, 7.78911455045350943659198408269, 8.659519045735364960050246875752, 9.409620230906231153855156709231, 10.67323897066444768780772321516