L(s) = 1 | + (0.866 + 1.5i)3-s + (−0.5 + 0.866i)5-s + (0.866 − 0.5i)7-s + (−1 + 1.73i)9-s − 1.73·15-s + (1.5 + 0.866i)21-s + (−0.866 + 1.5i)23-s + (−0.499 − 0.866i)25-s − 1.73·27-s + 29-s + 0.999i·35-s + 41-s − 1.73·43-s + (−1 − 1.73i)45-s + (0.499 − 0.866i)49-s + ⋯ |
L(s) = 1 | + (0.866 + 1.5i)3-s + (−0.5 + 0.866i)5-s + (0.866 − 0.5i)7-s + (−1 + 1.73i)9-s − 1.73·15-s + (1.5 + 0.866i)21-s + (−0.866 + 1.5i)23-s + (−0.499 − 0.866i)25-s − 1.73·27-s + 29-s + 0.999i·35-s + 41-s − 1.73·43-s + (−1 − 1.73i)45-s + (0.499 − 0.866i)49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.553 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.553 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.508222108\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.508222108\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.5 - 0.866i)T \) |
| 7 | \( 1 + (-0.866 + 0.5i)T \) |
good | 3 | \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 - T + T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 - T + T^{2} \) |
| 43 | \( 1 + 1.73T + T^{2} \) |
| 47 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 - 1.73T + T^{2} \) |
| 89 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.622397850171541925655973331182, −8.712265794360440474014174748398, −7.968128670633365355090291440202, −7.54539077335710462413053001828, −6.38373625051062535054015521722, −5.21017691248639200814095168590, −4.48614914201934789342961272042, −3.73590462632019633247008068801, −3.15086708779710540534563318787, −2.00700563404280842682165738803,
0.995465063009623840943227729977, 1.97403236975498787247187307384, 2.79297657836811247822203636972, 4.05595701494413920159529219102, 4.94811284309135931179561402270, 5.97748033828929447693342576953, 6.79583859394006231018577413583, 7.66028082875664684628393081876, 8.284711421416115061303373179080, 8.534669255616383657597846182557