L(s) = 1 | + i·2-s − 2.44i·3-s − 4-s + (2.22 + 0.224i)5-s + 2.44·6-s + i·7-s − i·8-s − 2.99·9-s + (−0.224 + 2.22i)10-s − 4.89·11-s + 2.44i·12-s + 4.44i·13-s − 14-s + (0.550 − 5.44i)15-s + 16-s − 2i·17-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 1.41i·3-s − 0.5·4-s + (0.994 + 0.100i)5-s + 0.999·6-s + 0.377i·7-s − 0.353i·8-s − 0.999·9-s + (−0.0710 + 0.703i)10-s − 1.47·11-s + 0.707i·12-s + 1.23i·13-s − 0.267·14-s + (0.142 − 1.40i)15-s + 0.250·16-s − 0.485i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.100i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 + 0.100i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.942687 - 0.0474944i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.942687 - 0.0474944i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 5 | \( 1 + (-2.22 - 0.224i)T \) |
| 7 | \( 1 - iT \) |
good | 3 | \( 1 + 2.44iT - 3T^{2} \) |
| 11 | \( 1 + 4.89T + 11T^{2} \) |
| 13 | \( 1 - 4.44iT - 13T^{2} \) |
| 17 | \( 1 + 2iT - 17T^{2} \) |
| 19 | \( 1 + 1.55T + 19T^{2} \) |
| 23 | \( 1 - 2.89iT - 23T^{2} \) |
| 29 | \( 1 + 6.89T + 29T^{2} \) |
| 31 | \( 1 - 8.89T + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 + 1.10T + 41T^{2} \) |
| 43 | \( 1 + 0.898iT - 43T^{2} \) |
| 47 | \( 1 + 8.89iT - 47T^{2} \) |
| 53 | \( 1 + 10.8iT - 53T^{2} \) |
| 59 | \( 1 - 1.55T + 59T^{2} \) |
| 61 | \( 1 - 3.55T + 61T^{2} \) |
| 67 | \( 1 - 8iT - 67T^{2} \) |
| 71 | \( 1 + 1.10T + 71T^{2} \) |
| 73 | \( 1 - 2.89iT - 73T^{2} \) |
| 79 | \( 1 + 6.89T + 79T^{2} \) |
| 83 | \( 1 + 2.44iT - 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 + 15.7iT - 97T^{2} \) |
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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
−14.44646370975234255480349180264, −13.45973863280713756712153726458, −13.01986328622609529127605007166, −11.66693394927099345644334706199, −9.972584520176235140764895497419, −8.648810880927592417393614859814, −7.41215864755701840767877718374, −6.44253967075169279419816373286, −5.29435469974098243385809056628, −2.20761612620777380177549525069,
2.90486681587310966934221475522, 4.61801799198574011265464084909, 5.69448156402581075408846634244, 8.163324242375110243183970953295, 9.522611812166983285514215208849, 10.43134582684158402341125059784, 10.73426123510157600035228413424, 12.69730178009787096880303935153, 13.50002272829615240033097414624, 14.80967854907105242526667714811