L(s) = 1 | + 2-s + 3.40i·3-s + 4-s + (1.28 − 1.83i)5-s + 3.40i·6-s + 2.06i·7-s + 8-s − 8.58·9-s + (1.28 − 1.83i)10-s + 3.77·11-s + 3.40i·12-s + 2.88·13-s + 2.06i·14-s + (6.23 + 4.36i)15-s + 16-s − 5.80·17-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.96i·3-s + 0.5·4-s + (0.573 − 0.819i)5-s + 1.38i·6-s + 0.779i·7-s + 0.353·8-s − 2.86·9-s + (0.405 − 0.579i)10-s + 1.13·11-s + 0.982i·12-s + 0.799·13-s + 0.551i·14-s + (1.60 + 1.12i)15-s + 0.250·16-s − 1.40·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0485 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0485 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.48952 + 1.56365i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.48952 + 1.56365i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 5 | \( 1 + (-1.28 + 1.83i)T \) |
| 37 | \( 1 + (-4.80 + 3.72i)T \) |
good | 3 | \( 1 - 3.40iT - 3T^{2} \) |
| 7 | \( 1 - 2.06iT - 7T^{2} \) |
| 11 | \( 1 - 3.77T + 11T^{2} \) |
| 13 | \( 1 - 2.88T + 13T^{2} \) |
| 17 | \( 1 + 5.80T + 17T^{2} \) |
| 19 | \( 1 + 0.157iT - 19T^{2} \) |
| 23 | \( 1 + 5.41T + 23T^{2} \) |
| 29 | \( 1 + 4.29iT - 29T^{2} \) |
| 31 | \( 1 - 0.425iT - 31T^{2} \) |
| 41 | \( 1 - 0.923T + 41T^{2} \) |
| 43 | \( 1 - 10.8T + 43T^{2} \) |
| 47 | \( 1 - 0.676iT - 47T^{2} \) |
| 53 | \( 1 - 9.87iT - 53T^{2} \) |
| 59 | \( 1 + 8.47iT - 59T^{2} \) |
| 61 | \( 1 - 1.23iT - 61T^{2} \) |
| 67 | \( 1 - 6.45iT - 67T^{2} \) |
| 71 | \( 1 + 3.28T + 71T^{2} \) |
| 73 | \( 1 + 0.980iT - 73T^{2} \) |
| 79 | \( 1 - 8.04iT - 79T^{2} \) |
| 83 | \( 1 + 11.9iT - 83T^{2} \) |
| 89 | \( 1 + 7.65iT - 89T^{2} \) |
| 97 | \( 1 + 13.9T + 97T^{2} \) |
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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
−11.51206057177961330035249218580, −10.81528343649301337968421816827, −9.666974568996047762055726795705, −9.090123877560904822866958140507, −8.422259398764974047163107905114, −6.11562431643264340614257543653, −5.72881392711964161739494265295, −4.43259981050831167238375387271, −4.03531936196426417872289382423, −2.45348188311889124321522534239,
1.40785865743634044270907486701, 2.51013778297215904582479656334, 3.87859067035351649990381746799, 5.81063244608675935986800453142, 6.54763927940053896543789579421, 6.94895235210604881102903591510, 7.979862017414054766759494578752, 9.135105624412960113319579335300, 10.75831657531488888754355709432, 11.35184783741575239770738636436