| L(s) = 1 | + (−1.5 − 2.59i)3-s + (−2.78 + 4.81i)5-s + (−9.67 − 15.7i)7-s + (−4.5 + 7.79i)9-s + (−6.95 − 12.0i)11-s + 38.6·13-s + 16.6·15-s + (−21.7 − 37.6i)17-s + (−54.5 + 94.4i)19-s + (−26.5 + 48.8i)21-s + (−37.4 + 64.8i)23-s + (47.0 + 81.4i)25-s + 27·27-s − 72.3·29-s + (32.0 + 55.4i)31-s + ⋯ |
| L(s) = 1 | + (−0.288 − 0.499i)3-s + (−0.248 + 0.430i)5-s + (−0.522 − 0.852i)7-s + (−0.166 + 0.288i)9-s + (−0.190 − 0.330i)11-s + 0.825·13-s + 0.287·15-s + (−0.310 − 0.537i)17-s + (−0.658 + 1.14i)19-s + (−0.275 + 0.507i)21-s + (−0.339 + 0.587i)23-s + (0.376 + 0.651i)25-s + 0.192·27-s − 0.463·29-s + (0.185 + 0.321i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0895 - 0.995i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0895 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.6590378633\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6590378633\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.5 + 2.59i)T \) |
| 7 | \( 1 + (9.67 + 15.7i)T \) |
| good | 5 | \( 1 + (2.78 - 4.81i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (6.95 + 12.0i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 - 38.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + (21.7 + 37.6i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (54.5 - 94.4i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (37.4 - 64.8i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + 72.3T + 2.43e4T^{2} \) |
| 31 | \( 1 + (-32.0 - 55.4i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (94.3 - 163. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + 24.7T + 6.89e4T^{2} \) |
| 43 | \( 1 - 243.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (310. - 537. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-143. - 249. i)T + (-7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-262. - 454. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-191. + 332. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-99.0 - 171. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + 785.T + 3.57e5T^{2} \) |
| 73 | \( 1 + (-165. - 286. i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-218. + 379. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + 241.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (792. - 1.37e3i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 - 79.2T + 9.12e5T^{2} \) |
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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
−11.20688780613976837161213571026, −10.67429223294265251279376228186, −9.630060970725683562671545545848, −8.374612135062131046058729170308, −7.44903746127152810965432689199, −6.60796308578834075987576766375, −5.70040558023288124257786437286, −4.14325044447980135400695694148, −3.08303153134362467620373442048, −1.30712247081533675948603369989,
0.25730605606593441529918929899, 2.30491151276552549928641510857, 3.77923173955743028313161391162, 4.84075060893001106812472135075, 5.92422547671456639206689908385, 6.81988569192304260011958078662, 8.416270253941297217876608329763, 8.893538203155908869568426180437, 9.973319265080125215654461466911, 10.88932870407074104202209239360
