L(s) = 1 | + (1.5 + 0.866i)3-s + (1.5 + 2.59i)9-s − 3·11-s − 2·13-s + 5.19i·17-s + 5.19i·19-s + 6·23-s + 5.19i·27-s + 10.3i·29-s − 3.46i·31-s + (−4.5 − 2.59i)33-s + 8·37-s + (−3 − 1.73i)39-s + 5.19i·41-s − 3.46i·43-s + ⋯ |
L(s) = 1 | + (0.866 + 0.499i)3-s + (0.5 + 0.866i)9-s − 0.904·11-s − 0.554·13-s + 1.26i·17-s + 1.19i·19-s + 1.25·23-s + 0.999i·27-s + 1.92i·29-s − 0.622i·31-s + (−0.783 − 0.452i)33-s + 1.31·37-s + (−0.480 − 0.277i)39-s + 0.811i·41-s − 0.528i·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.921145275\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.921145275\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.5 - 0.866i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 + 3T + 11T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 - 5.19iT - 17T^{2} \) |
| 19 | \( 1 - 5.19iT - 19T^{2} \) |
| 23 | \( 1 - 6T + 23T^{2} \) |
| 29 | \( 1 - 10.3iT - 29T^{2} \) |
| 31 | \( 1 + 3.46iT - 31T^{2} \) |
| 37 | \( 1 - 8T + 37T^{2} \) |
| 41 | \( 1 - 5.19iT - 41T^{2} \) |
| 43 | \( 1 + 3.46iT - 43T^{2} \) |
| 47 | \( 1 + 6T + 47T^{2} \) |
| 53 | \( 1 + 10.3iT - 53T^{2} \) |
| 59 | \( 1 + 12T + 59T^{2} \) |
| 61 | \( 1 - 8T + 61T^{2} \) |
| 67 | \( 1 + 12.1iT - 67T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 - T + 73T^{2} \) |
| 79 | \( 1 - 6.92iT - 79T^{2} \) |
| 83 | \( 1 + 9T + 83T^{2} \) |
| 89 | \( 1 - 5.19iT - 89T^{2} \) |
| 97 | \( 1 - 10T + 97T^{2} \) |
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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
−9.949071215160877709405285288689, −9.148258902572960841846009259247, −8.255774497202089660090663603470, −7.77666664841938190662447383816, −6.77560219337028756121239169548, −5.54815261604095958672505092947, −4.76447631261619758628405568919, −3.71331411593505908329064024068, −2.85753585158968917578674971710, −1.70698974645208637812633518352,
0.73878341732076462904245968158, 2.48302961494227328366943154877, 2.88253801903961845956274903295, 4.34226208320684842695419885883, 5.18343743253181781866862967579, 6.40687562695412033412906492299, 7.34009431470494632789417767851, 7.71661740425551270066394270788, 8.807714535280942665412841301694, 9.388712944607297374453685683482