L(s) = 1 | − 3i·3-s + 5.86·5-s + (−18.3 − 2.64i)7-s − 9·9-s + 34.5·11-s + 55.8·13-s − 17.5i·15-s + 2.77i·17-s + 67.8i·19-s + (−7.94 + 54.9i)21-s + 176. i·23-s − 90.6·25-s + 27i·27-s − 116. i·29-s − 312.·31-s + ⋯ |
L(s) = 1 | − 0.577i·3-s + 0.524·5-s + (−0.989 − 0.143i)7-s − 0.333·9-s + 0.946·11-s + 1.19·13-s − 0.302i·15-s + 0.0395i·17-s + 0.819i·19-s + (−0.0826 + 0.571i)21-s + 1.59i·23-s − 0.724·25-s + 0.192i·27-s − 0.743i·29-s − 1.80·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.394 - 0.918i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.394 - 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.487913455\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.487913455\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + 3iT \) |
| 7 | \( 1 + (18.3 + 2.64i)T \) |
good | 5 | \( 1 - 5.86T + 125T^{2} \) |
| 11 | \( 1 - 34.5T + 1.33e3T^{2} \) |
| 13 | \( 1 - 55.8T + 2.19e3T^{2} \) |
| 17 | \( 1 - 2.77iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 67.8iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 176. iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 116. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 312.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 118. iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 280. iT - 6.89e4T^{2} \) |
| 43 | \( 1 - 15.1T + 7.95e4T^{2} \) |
| 47 | \( 1 + 6.34T + 1.03e5T^{2} \) |
| 53 | \( 1 - 23.2iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 288. iT - 2.05e5T^{2} \) |
| 61 | \( 1 + 514.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 295.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 475. iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 473. iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 796. iT - 4.93e5T^{2} \) |
| 83 | \( 1 - 877. iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 33.8iT - 7.04e5T^{2} \) |
| 97 | \( 1 + 700. iT - 9.12e5T^{2} \) |
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
−9.376256248671565468156924164088, −8.710148978311649455024727030303, −7.63776142660599079110531612761, −6.89965816365539519855230050913, −5.95208777584569852510455783543, −5.72025931426662082426798194883, −3.93022701714768195988718705945, −3.42473170410419989632640061130, −1.97545674608570855948309360605, −1.12955984260438886323236807223,
0.35605680640433183510891238600, 1.79430077205299728648797933433, 3.07520647400009134175236969006, 3.81707701846976045287095493134, 4.80275928887277779126750889988, 6.02042472658441675336831409292, 6.32412251799233412421084065314, 7.34635449976660200290654857734, 8.739587047186676818633816271816, 9.044728751939392973072805622746