L(s) = 1 | + (1.38 − 1.38i)2-s − 2.84i·4-s + (0.923 + 0.382i)5-s + (−0.360 + 0.149i)7-s + (−2.56 − 2.56i)8-s + (−0.707 − 0.707i)9-s + (1.81 − 0.750i)10-s + (0.382 + 0.923i)11-s + 1.11i·13-s + (−0.292 + 0.707i)14-s − 4.26·16-s + (−0.555 + 0.831i)17-s − 1.96·18-s + (1.08 − 2.63i)20-s + (1.81 + 0.750i)22-s + ⋯ |
L(s) = 1 | + (1.38 − 1.38i)2-s − 2.84i·4-s + (0.923 + 0.382i)5-s + (−0.360 + 0.149i)7-s + (−2.56 − 2.56i)8-s + (−0.707 − 0.707i)9-s + (1.81 − 0.750i)10-s + (0.382 + 0.923i)11-s + 1.11i·13-s + (−0.292 + 0.707i)14-s − 4.26·16-s + (−0.555 + 0.831i)17-s − 1.96·18-s + (1.08 − 2.63i)20-s + (1.81 + 0.750i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 935 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.490 + 0.871i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 935 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.490 + 0.871i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.046329137\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.046329137\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-0.923 - 0.382i)T \) |
| 11 | \( 1 + (-0.382 - 0.923i)T \) |
| 17 | \( 1 + (0.555 - 0.831i)T \) |
good | 2 | \( 1 + (-1.38 + 1.38i)T - iT^{2} \) |
| 3 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 7 | \( 1 + (0.360 - 0.149i)T + (0.707 - 0.707i)T^{2} \) |
| 13 | \( 1 - 1.11iT - T^{2} \) |
| 19 | \( 1 + iT^{2} \) |
| 23 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 29 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 31 | \( 1 + (-0.541 + 1.30i)T + (-0.707 - 0.707i)T^{2} \) |
| 37 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 41 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 43 | \( 1 + (1.38 + 1.38i)T + iT^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + iT^{2} \) |
| 59 | \( 1 + iT^{2} \) |
| 61 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + (0.292 - 0.707i)T + (-0.707 - 0.707i)T^{2} \) |
| 73 | \( 1 + (-1.53 - 0.636i)T + (0.707 + 0.707i)T^{2} \) |
| 79 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 83 | \( 1 + (-1.17 + 1.17i)T - iT^{2} \) |
| 89 | \( 1 + 0.765iT - T^{2} \) |
| 97 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
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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
−10.10541354608870920493960314724, −9.550206594246615627429617403882, −8.916201992507628866023598169634, −6.70876615756006679324684454546, −6.36162688293192556039256469925, −5.48117792534292638808184882749, −4.41721273504660288933990093376, −3.58396732990695429780837594194, −2.47739684351878657863758756924, −1.71998365660705014530523070954,
2.67510455642308582944582763121, 3.40689655816509096934703356692, 4.89234498234464172676189997088, 5.26801164220346074506058068161, 6.19681266292221314739529579182, 6.72056372115783056891683513012, 7.969797302011231717142913766909, 8.468844446736348534984719557327, 9.358050564838809487524440054082, 10.70365473342931551921869893767