L(s) = 1 | + i·3-s + 2.64·7-s − 9-s + 3.58i·11-s − 7.02i·13-s − 7.58·17-s − 4.58i·19-s + 2.64i·21-s + 2.55·23-s − i·27-s + 4.37i·29-s − 4.28·31-s − 3.58·33-s + 6.92i·37-s + 7.02·39-s + ⋯ |
L(s) = 1 | + 0.577i·3-s + 0.999·7-s − 0.333·9-s + 1.08i·11-s − 1.94i·13-s − 1.83·17-s − 1.05i·19-s + 0.577i·21-s + 0.531·23-s − 0.192i·27-s + 0.812i·29-s − 0.769·31-s − 0.623·33-s + 1.13i·37-s + 1.12·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 - 0.258i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.965 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6510957686\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6510957686\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 2.64T + 7T^{2} \) |
| 11 | \( 1 - 3.58iT - 11T^{2} \) |
| 13 | \( 1 + 7.02iT - 13T^{2} \) |
| 17 | \( 1 + 7.58T + 17T^{2} \) |
| 19 | \( 1 + 4.58iT - 19T^{2} \) |
| 23 | \( 1 - 2.55T + 23T^{2} \) |
| 29 | \( 1 - 4.37iT - 29T^{2} \) |
| 31 | \( 1 + 4.28T + 31T^{2} \) |
| 37 | \( 1 - 6.92iT - 37T^{2} \) |
| 41 | \( 1 + 9.58T + 41T^{2} \) |
| 43 | \( 1 - 4.58iT - 43T^{2} \) |
| 47 | \( 1 + 5.29T + 47T^{2} \) |
| 53 | \( 1 - 9.66iT - 53T^{2} \) |
| 59 | \( 1 - 9.16iT - 59T^{2} \) |
| 61 | \( 1 + 0.0953iT - 61T^{2} \) |
| 67 | \( 1 - 4.58iT - 67T^{2} \) |
| 71 | \( 1 + 16.5T + 71T^{2} \) |
| 73 | \( 1 - 2T + 73T^{2} \) |
| 79 | \( 1 + 5.29T + 79T^{2} \) |
| 83 | \( 1 - 10iT - 83T^{2} \) |
| 89 | \( 1 - 3.16T + 89T^{2} \) |
| 97 | \( 1 - 0.165T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.713187285073999658210622689481, −7.925811002111152893134129543440, −7.23999077364079405966312957798, −6.50603581426460826349334706214, −5.40991058064202440704461234038, −4.84306023154454685984979344723, −4.43345126172840755650497325688, −3.21326900651970180551824604018, −2.47847254915169669223811482318, −1.35949032589839098014345783545,
0.16466130082308467060908890921, 1.79326696689590188261316011172, 1.97005470247618433983427452271, 3.41144449893264396911215854738, 4.25826878266539885460429891813, 4.94617446581972119880375515781, 5.90616489789131290194433094273, 6.57523777889421728020411415373, 7.14951793708483465690774168120, 8.009848268481883818016694994958