L(s) = 1 | + 1.20i·2-s + 0.539·4-s + (−1.52 + 1.52i)5-s + (0.707 + 0.707i)7-s + 3.06i·8-s + (−1.84 − 1.84i)10-s + (−4.46 − 4.46i)11-s − 5.08·13-s + (−0.854 + 0.854i)14-s − 2.63·16-s + (−4.00 − 0.968i)17-s − 7.96i·19-s + (−0.824 + 0.824i)20-s + (5.39 − 5.39i)22-s + (1.06 + 1.06i)23-s + ⋯ |
L(s) = 1 | + 0.854i·2-s + 0.269·4-s + (−0.683 + 0.683i)5-s + (0.267 + 0.267i)7-s + 1.08i·8-s + (−0.584 − 0.584i)10-s + (−1.34 − 1.34i)11-s − 1.41·13-s + (−0.228 + 0.228i)14-s − 0.657·16-s + (−0.972 − 0.234i)17-s − 1.82i·19-s + (−0.184 + 0.184i)20-s + (1.15 − 1.15i)22-s + (0.222 + 0.222i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1071 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.351 + 0.936i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1071 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.351 + 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1387436535\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1387436535\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (-0.707 - 0.707i)T \) |
| 17 | \( 1 + (4.00 + 0.968i)T \) |
good | 2 | \( 1 - 1.20iT - 2T^{2} \) |
| 5 | \( 1 + (1.52 - 1.52i)T - 5iT^{2} \) |
| 11 | \( 1 + (4.46 + 4.46i)T + 11iT^{2} \) |
| 13 | \( 1 + 5.08T + 13T^{2} \) |
| 19 | \( 1 + 7.96iT - 19T^{2} \) |
| 23 | \( 1 + (-1.06 - 1.06i)T + 23iT^{2} \) |
| 29 | \( 1 + (2.62 - 2.62i)T - 29iT^{2} \) |
| 31 | \( 1 + (-3.12 + 3.12i)T - 31iT^{2} \) |
| 37 | \( 1 + (5.76 - 5.76i)T - 37iT^{2} \) |
| 41 | \( 1 + (-5.51 - 5.51i)T + 41iT^{2} \) |
| 43 | \( 1 - 7.20iT - 43T^{2} \) |
| 47 | \( 1 - 1.30T + 47T^{2} \) |
| 53 | \( 1 + 6.85iT - 53T^{2} \) |
| 59 | \( 1 + 12.2iT - 59T^{2} \) |
| 61 | \( 1 + (3.53 + 3.53i)T + 61iT^{2} \) |
| 67 | \( 1 + 0.226T + 67T^{2} \) |
| 71 | \( 1 + (0.416 - 0.416i)T - 71iT^{2} \) |
| 73 | \( 1 + (9.39 - 9.39i)T - 73iT^{2} \) |
| 79 | \( 1 + (-3.12 - 3.12i)T + 79iT^{2} \) |
| 83 | \( 1 + 2.55iT - 83T^{2} \) |
| 89 | \( 1 + 10.8T + 89T^{2} \) |
| 97 | \( 1 + (9.49 - 9.49i)T - 97iT^{2} \) |
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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
−10.74144150884940168981157280778, −9.477838135029890032659920925813, −8.464903073356388038559554644820, −7.83227432138008737307658542898, −7.13476744042061179314583129909, −6.48654981993335073677894737740, −5.32173498096776185228665281227, −4.77684825051265010539799957202, −2.99842052891849737840760913209, −2.51935236231016852566843000166,
0.05478373852315421997190796934, 1.81309400592031309888761839944, 2.60012915769198763672759194912, 4.05319567563295553225301538492, 4.60471077071146526913376623896, 5.70591271182332653316135476610, 7.25499000234051819499481049868, 7.45428097710188079662185204990, 8.503087861023808160303157012846, 9.604811713874592006931906790000