L(s) = 1 | + (−0.541 − 0.541i)3-s + i·4-s + (−0.923 − 0.382i)5-s − 0.414i·9-s + 11-s + (0.541 − 0.541i)12-s + (0.292 + 0.707i)15-s − 16-s + (0.382 − 0.923i)20-s + (0.707 + 0.707i)25-s + (−0.765 + 0.765i)27-s − 0.765i·31-s + (−0.541 − 0.541i)33-s + 0.414·36-s + (−1 − i)37-s + ⋯ |
L(s) = 1 | + (−0.541 − 0.541i)3-s + i·4-s + (−0.923 − 0.382i)5-s − 0.414i·9-s + 11-s + (0.541 − 0.541i)12-s + (0.292 + 0.707i)15-s − 16-s + (0.382 − 0.923i)20-s + (0.707 + 0.707i)25-s + (−0.765 + 0.765i)27-s − 0.765i·31-s + (−0.541 − 0.541i)33-s + 0.414·36-s + (−1 − i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2695 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.549 + 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2695 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.549 + 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7831114165\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7831114165\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (0.923 + 0.382i)T \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
good | 2 | \( 1 - iT^{2} \) |
| 3 | \( 1 + (0.541 + 0.541i)T + iT^{2} \) |
| 13 | \( 1 - iT^{2} \) |
| 17 | \( 1 + iT^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - iT^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 + 0.765iT - T^{2} \) |
| 37 | \( 1 + (1 + i)T + iT^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 47 | \( 1 + (-1.30 + 1.30i)T - iT^{2} \) |
| 53 | \( 1 + (-1.41 + 1.41i)T - iT^{2} \) |
| 59 | \( 1 - 1.84T + T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 + (1.41 + 1.41i)T + iT^{2} \) |
| 71 | \( 1 - 1.41T + T^{2} \) |
| 73 | \( 1 - iT^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 - iT^{2} \) |
| 89 | \( 1 - 0.765T + T^{2} \) |
| 97 | \( 1 + (-1.30 + 1.30i)T - iT^{2} \) |
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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
−8.822782436559724133825828040441, −8.124739972028655765607202834022, −7.17690816127142909568062255466, −6.96338766199589428464599832941, −5.94303759328118593313558499795, −4.92953745135474099315826862762, −3.74343198357367632837731296850, −3.70843333203486961082356853250, −2.12518671453296376385964816590, −0.65148564708925073594478721935,
1.14493126819474428772790972595, 2.51204201488154730844977811619, 3.79709261951104349019791321619, 4.46236826035861004355170214099, 5.21010820713123245423221712308, 6.02541173257438844321099711561, 6.79491145926447764683391492566, 7.47058828481620615165567331589, 8.545017068767187893133630167509, 9.160644570047581590475061664390