L(s) = 1 | + (−1 − i)13-s + (−1.41 − 1.41i)17-s − 1.41·29-s + (1 − i)37-s − 1.41i·41-s − i·49-s + (−1.41 + 1.41i)53-s + (1 + i)73-s + 1.41·89-s + (1 − i)97-s − 1.41i·101-s + ⋯ |
L(s) = 1 | + (−1 − i)13-s + (−1.41 − 1.41i)17-s − 1.41·29-s + (1 − i)37-s − 1.41i·41-s − i·49-s + (−1.41 + 1.41i)53-s + (1 + i)73-s + 1.41·89-s + (1 − i)97-s − 1.41i·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.391 + 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.391 + 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7407791707\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7407791707\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + iT^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 + (1 + i)T + iT^{2} \) |
| 17 | \( 1 + (1.41 + 1.41i)T + iT^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + iT^{2} \) |
| 29 | \( 1 + 1.41T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + (-1 + i)T - iT^{2} \) |
| 41 | \( 1 + 1.41iT - T^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 47 | \( 1 - iT^{2} \) |
| 53 | \( 1 + (1.41 - 1.41i)T - iT^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 + iT^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + (-1 - i)T + iT^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 + iT^{2} \) |
| 89 | \( 1 - 1.41T + T^{2} \) |
| 97 | \( 1 + (-1 + i)T - iT^{2} \) |
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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
−8.567248892908303044708224629718, −7.46839342528560548135297605897, −7.32181432205187540716255600752, −6.25443319809882618780658313562, −5.39124956076113367634690634621, −4.78952659669949514086963591717, −3.86226711840103339051260172007, −2.79469070822809598425280377879, −2.10734300108169582526941406054, −0.39850318103113549641196065899,
1.65719099457495338257020987190, 2.41360974141554376186838932514, 3.60535296727699148717555715829, 4.43822268827050100980762420462, 5.01615814203792437299528726007, 6.29596244471888632877575886245, 6.50876609373685622225389472634, 7.58603639358810626738074282252, 8.142141148309820052391565959894, 9.103922071014337572811379018402