L(s) = 1 | − 1.41i·2-s − 2.00·4-s + 2.82i·8-s + 4.00·16-s + 7.87i·17-s − 5·25-s + 7.87i·29-s + 5.56·31-s − 5.65i·32-s + 11.1·34-s + 11.1·43-s + 1.41i·47-s + 7·49-s + 7.07i·50-s + 7.87i·53-s + ⋯ |
L(s) = 1 | − 0.999i·2-s − 1.00·4-s + 1.00i·8-s + 1.00·16-s + 1.90i·17-s − 25-s + 1.46i·29-s + 1.00·31-s − 1.00i·32-s + 1.90·34-s + 1.69·43-s + 0.206i·47-s + 49-s + 0.999i·50-s + 1.08i·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1116 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1116 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.209199588\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.209199588\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 1.41iT \) |
| 3 | \( 1 \) |
| 31 | \( 1 - 5.56T \) |
good | 5 | \( 1 + 5T^{2} \) |
| 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 7.87iT - 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 7.87iT - 29T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 11.1T + 43T^{2} \) |
| 47 | \( 1 - 1.41iT - 47T^{2} \) |
| 53 | \( 1 - 7.87iT - 53T^{2} \) |
| 59 | \( 1 + 7.07iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 - 9.89iT - 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 - 11.1T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 - 7.87iT - 89T^{2} \) |
| 97 | \( 1 + 4T + 97T^{2} \) |
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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
−10.02823086390321757138093194968, −9.105389504212687051384678172177, −8.406299928352117171290119377650, −7.61274329546394160311311357655, −6.28212237291547466587029169893, −5.46429230084732328308210094454, −4.32147862298709959483989177862, −3.61444013622569900978559618053, −2.41349817698543257459087489629, −1.28090889499958778622894263624,
0.60374756455491515360145278190, 2.58546694277605090493116186104, 3.92307044648178206620584028391, 4.79358789200637832067230103975, 5.67494631280870866605629412617, 6.49988457234966172836817629412, 7.42773096387680199117310229885, 7.950682778215771256725579835931, 9.030467534785756562020024787007, 9.583354941250212472227692670374