L(s) = 1 | − 16.7i·5-s + 22.3·7-s + 51.9i·11-s − 28.0i·13-s − 60.9·17-s + 162. i·19-s − 83.2·23-s − 156.·25-s − 217. i·29-s + 174.·31-s − 374. i·35-s − 407. i·37-s + 196.·41-s − 345. i·43-s − 335.·47-s + ⋯ |
L(s) = 1 | − 1.50i·5-s + 1.20·7-s + 1.42i·11-s − 0.598i·13-s − 0.869·17-s + 1.96i·19-s − 0.755·23-s − 1.25·25-s − 1.39i·29-s + 1.00·31-s − 1.80i·35-s − 1.81i·37-s + 0.748·41-s − 1.22i·43-s − 1.04·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.568820488\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.568820488\) |
\(L(\frac{5}{2})\) |
|
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 | \( 1 + 16.7iT - 125T^{2} \) |
| 7 | \( 1 - 22.3T + 343T^{2} \) |
| 11 | \( 1 - 51.9iT - 1.33e3T^{2} \) |
| 13 | \( 1 + 28.0iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 60.9T + 4.91e3T^{2} \) |
| 19 | \( 1 - 162. iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 83.2T + 1.21e4T^{2} \) |
| 29 | \( 1 + 217. iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 174.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 407. iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 196.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 345. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 335.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 234. iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 321. iT - 2.05e5T^{2} \) |
| 61 | \( 1 + 739. iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 20iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 418.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 397.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 95.6T + 4.93e5T^{2} \) |
| 83 | \( 1 + 348. iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 1.49e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 881.T + 9.12e5T^{2} \) |
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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
−8.371873047859299910574383365648, −8.133966252131542695963765584616, −7.34269892797669480424662482124, −6.00035676870645904061391680198, −5.29485838048465511551657735609, −4.46057844846389312663936233206, −4.04617159936699514143749145994, −2.10876787636461743366480871122, −1.59773549971785956342591427238, −0.33084168405482716410499986269,
1.21353645359614542259784971800, 2.51761089074934207145791874072, 3.09856868487461781085138774510, 4.32759097917518276940869523885, 5.10520769293795865459731224702, 6.38341926413017309614126204624, 6.67126984778837477398765968903, 7.67709510499580151410599949264, 8.440829761449917109405645185963, 9.118748771422194369204924786171