| 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
−9.388712944607297374453685683482, −8.807714535280942665412841301694, −7.71661740425551270066394270788, −7.34009431470494632789417767851, −6.40687562695412033412906492299, −5.18343743253181781866862967579, −4.34226208320684842695419885883, −2.88253801903961845956274903295, −2.48302961494227328366943154877, −0.73878341732076462904245968158,
1.70698974645208637812633518352, 2.85753585158968917578674971710, 3.71331411593505908329064024068, 4.76447631261619758628405568919, 5.54815261604095958672505092947, 6.77560219337028756121239169548, 7.77666664841938190662447383816, 8.255774497202089660090663603470, 9.148258902572960841846009259247, 9.949071215160877709405285288689
