L(s) = 1 | − 0.378i·7-s + (2.46 − 2.46i)13-s + (−2.44 + 2.44i)19-s − 5i·25-s + 10.1·31-s + (−1.53 − 1.53i)37-s + (−7.34 − 7.34i)43-s + 6.85·49-s + (10.4 − 10.4i)61-s + (−11.3 + 11.3i)67-s − 13.8i·73-s + 9.41·79-s + (−0.933 − 0.933i)91-s + 13.8·97-s − 11.6i·103-s + ⋯ |
L(s) = 1 | − 0.143i·7-s + (0.683 − 0.683i)13-s + (−0.561 + 0.561i)19-s − i·25-s + 1.82·31-s + (−0.252 − 0.252i)37-s + (−1.12 − 1.12i)43-s + 0.979·49-s + (1.33 − 1.33i)61-s + (−1.38 + 1.38i)67-s − 1.62i·73-s + 1.05·79-s + (−0.0978 − 0.0978i)91-s + 1.40·97-s − 1.15i·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.608 + 0.793i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.608 + 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.673238633\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.673238633\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 5iT^{2} \) |
| 7 | \( 1 + 0.378iT - 7T^{2} \) |
| 11 | \( 1 + 11iT^{2} \) |
| 13 | \( 1 + (-2.46 + 2.46i)T - 13iT^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + (2.44 - 2.44i)T - 19iT^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 - 29iT^{2} \) |
| 31 | \( 1 - 10.1T + 31T^{2} \) |
| 37 | \( 1 + (1.53 + 1.53i)T + 37iT^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + (7.34 + 7.34i)T + 43iT^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 53iT^{2} \) |
| 59 | \( 1 + 59iT^{2} \) |
| 61 | \( 1 + (-10.4 + 10.4i)T - 61iT^{2} \) |
| 67 | \( 1 + (11.3 - 11.3i)T - 67iT^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + 13.8iT - 73T^{2} \) |
| 79 | \( 1 - 9.41T + 79T^{2} \) |
| 83 | \( 1 - 83iT^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 - 13.8T + 97T^{2} \) |
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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
−8.590837745990840251315846044981, −8.348459692303878116852937088923, −7.38189610048997701421128095019, −6.46863431824450143063775973974, −5.87199538877238533175041721992, −4.87626102572588804044147141730, −4.00746745337212722744222798735, −3.11695992451155046724732611688, −2.00625974082682087726656183008, −0.65866564701460171188019754269,
1.14062921116334031518339444598, 2.32945836844049919345298260651, 3.36356506937563210708842703101, 4.32518828452182280318247243747, 5.08887604067039806158722968139, 6.14457270238975532315349091636, 6.68712298869376456418782165754, 7.58689085608942825619363266401, 8.498541613607827328473377871966, 8.980454838758594229273635865204