L(s) = 1 | + 3.14i·3-s − 6.89·9-s − 6.61i·11-s + 7.89·17-s + 2.51i·19-s − 12.2i·27-s + 20.7·33-s + 12.7·41-s + 8.48i·43-s − 7·49-s + 24.8i·51-s − 7.89·57-s + 14.1i·59-s + 7.88i·67-s + 13.6·73-s + ⋯ |
L(s) = 1 | + 1.81i·3-s − 2.29·9-s − 1.99i·11-s + 1.91·17-s + 0.575i·19-s − 2.36i·27-s + 3.62·33-s + 1.99·41-s + 1.29i·43-s − 49-s + 3.48i·51-s − 1.04·57-s + 1.84i·59-s + 0.962i·67-s + 1.60·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.800744994\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.800744994\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 - 3.14iT - 3T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 6.61iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 7.89T + 17T^{2} \) |
| 19 | \( 1 - 2.51iT - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 - 12.7T + 41T^{2} \) |
| 43 | \( 1 - 8.48iT - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 - 14.1iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 7.88iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 13.6T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 14.1iT - 83T^{2} \) |
| 89 | \( 1 - 13.8T + 89T^{2} \) |
| 97 | \( 1 - 10T + 97T^{2} \) |
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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
−9.002843042137891151576916708334, −8.256969570113695027632222404235, −7.70579061144435125041440390581, −6.03596950952434510151763251639, −5.83167351959017090698842848675, −5.02204259149807134998799818029, −4.06055528157955106569751127146, −3.37632961412445304443922075408, −2.88523792875482247434817285879, −0.888759081092566096352428329528,
0.76459456404511481518675418543, 1.78470301512211342053828682831, 2.43551730141395238827047347628, 3.52906941042186193753182432290, 4.83517668010488866546000748035, 5.56798810906579322176147291580, 6.49280077836508054166951406656, 7.04798387666760309235503187132, 7.74407200669558399974664496804, 7.994516776260248886622278416031