L(s) = 1 | + (−1.30 + 0.541i)2-s + (0.541 − 1.64i)3-s + (1.41 − 1.41i)4-s + (−1.25 − 1.84i)5-s + (0.183 + 2.44i)6-s − 3.29·7-s + (−1.08 + 2.61i)8-s + (−2.41 − 1.78i)9-s + (2.64 + 1.73i)10-s − 2.51i·11-s + (−1.56 − 3.09i)12-s + 4.65·13-s + (4.29 − 1.78i)14-s + (−3.72 + 1.07i)15-s − 4i·16-s + 3.69·17-s + ⋯ |
L(s) = 1 | + (−0.923 + 0.382i)2-s + (0.312 − 0.949i)3-s + (0.707 − 0.707i)4-s + (−0.563 − 0.826i)5-s + (0.0748 + 0.997i)6-s − 1.24·7-s + (−0.382 + 0.923i)8-s + (−0.804 − 0.593i)9-s + (0.836 + 0.547i)10-s − 0.759i·11-s + (−0.450 − 0.892i)12-s + 1.29·13-s + (1.14 − 0.475i)14-s + (−0.960 + 0.276i)15-s − i·16-s + 0.896·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.112 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.112 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.408894 - 0.457584i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.408894 - 0.457584i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.30 - 0.541i)T \) |
| 3 | \( 1 + (-0.541 + 1.64i)T \) |
| 5 | \( 1 + (1.25 + 1.84i)T \) |
good | 7 | \( 1 + 3.29T + 7T^{2} \) |
| 11 | \( 1 + 2.51iT - 11T^{2} \) |
| 13 | \( 1 - 4.65T + 13T^{2} \) |
| 17 | \( 1 - 3.69T + 17T^{2} \) |
| 19 | \( 1 - 0.828T + 19T^{2} \) |
| 23 | \( 1 + 2.61iT - 23T^{2} \) |
| 29 | \( 1 - 6.08T + 29T^{2} \) |
| 31 | \( 1 + 1.17iT - 31T^{2} \) |
| 37 | \( 1 - 1.92T + 37T^{2} \) |
| 41 | \( 1 - 8.59iT - 41T^{2} \) |
| 43 | \( 1 + 6.01iT - 43T^{2} \) |
| 47 | \( 1 - 2.61iT - 47T^{2} \) |
| 53 | \( 1 + 4.59iT - 53T^{2} \) |
| 59 | \( 1 - 2.51iT - 59T^{2} \) |
| 61 | \( 1 + 8.48iT - 61T^{2} \) |
| 67 | \( 1 - 3.29iT - 67T^{2} \) |
| 71 | \( 1 - 7.12T + 71T^{2} \) |
| 73 | \( 1 + 6.58iT - 73T^{2} \) |
| 79 | \( 1 - 16.4iT - 79T^{2} \) |
| 83 | \( 1 + 9.37T + 83T^{2} \) |
| 89 | \( 1 + 5.03iT - 89T^{2} \) |
| 97 | \( 1 - 2.72iT - 97T^{2} \) |
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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
−13.15308067929883444907106558607, −12.20116637610163682661251364066, −11.17528248020064573366001684043, −9.698826031970218220265434518231, −8.648746066818460185825467679494, −8.035279295185481705588386758815, −6.68867888071504281896100428092, −5.81069524151624465652302647658, −3.24691575059164648916771440846, −0.900382247523387602371549213780,
2.97194048793232155159349602487, 3.82837918064377410292606390095, 6.21997919624154830345733367663, 7.46318497893329638135211601259, 8.655969790533462698900268322225, 9.775917967362316787103410627326, 10.36534437260331951469417653580, 11.35618522044062298508909402618, 12.43375405022231617266504554302, 13.80434361386773242921087973744