L(s) = 1 | − 10.3i·5-s + 21.4·7-s + 24.7i·11-s + 21.4i·13-s + 123.·17-s + 7i·19-s − 114.·23-s + 17·25-s + 270. i·29-s + 214.·31-s − 222. i·35-s + 235. i·37-s − 395.·41-s + 92i·43-s − 114.·47-s + ⋯ |
L(s) = 1 | − 0.929i·5-s + 1.15·7-s + 0.678i·11-s + 0.457i·13-s + 1.76·17-s + 0.0845i·19-s − 1.03·23-s + 0.136·25-s + 1.73i·29-s + 1.24·31-s − 1.07i·35-s + 1.04i·37-s − 1.50·41-s + 0.326i·43-s − 0.354·47-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
\(L(2)\) |
\(\approx\) |
\(2.463787711\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.463787711\) |
\(L(\frac{5}{2})\) |
|
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 | \( 1 + 10.3iT - 125T^{2} \) |
| 7 | \( 1 - 21.4T + 343T^{2} \) |
| 11 | \( 1 - 24.7iT - 1.33e3T^{2} \) |
| 13 | \( 1 - 21.4iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 123.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 7iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 114.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 270. iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 214.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 235. iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 395.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 92iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 114.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 20.7iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 173. iT - 2.05e5T^{2} \) |
| 61 | \( 1 - 449. iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 353iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 789.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 425T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.30e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 593. iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 74.2T + 7.04e5T^{2} \) |
| 97 | \( 1 - 799T + 9.12e5T^{2} \) |
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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
−8.872824517131501525580947958869, −8.295539507328080779460775773974, −7.66617710682734737186488038980, −6.74153595113422873338316761395, −5.56967427210206488974956589244, −4.93229646988008318375897928849, −4.31904486131193613801416766949, −3.10562693431135813340551063842, −1.65922213300040722744285006228, −1.16743160874102306812327924721,
0.56173951121881049665300707345, 1.78298166232280878960155296582, 2.88258178676077067411798557267, 3.68702293632447373700618757431, 4.79381778921372225237872748323, 5.70580021104072622488153430114, 6.34480144998134202085043996946, 7.52113008143464337291538790011, 7.952618421133322054646782513791, 8.637071759813591185624900433274