| L(s) = 1 | + 3i·3-s + 4i·7-s − 9·9-s − 72·11-s − 6i·13-s − 38i·17-s + 52·19-s − 12·21-s − 152i·23-s − 27i·27-s + 78·29-s − 120·31-s − 216i·33-s + 150i·37-s + 18·39-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + 0.215i·7-s − 0.333·9-s − 1.97·11-s − 0.128i·13-s − 0.542i·17-s + 0.627·19-s − 0.124·21-s − 1.37i·23-s − 0.192i·27-s + 0.499·29-s − 0.695·31-s − 1.13i·33-s + 0.666i·37-s + 0.0739·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.504756912\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.504756912\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 3iT \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 4iT - 343T^{2} \) |
| 11 | \( 1 + 72T + 1.33e3T^{2} \) |
| 13 | \( 1 + 6iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 38iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 52T + 6.85e3T^{2} \) |
| 23 | \( 1 + 152iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 78T + 2.43e4T^{2} \) |
| 31 | \( 1 + 120T + 2.97e4T^{2} \) |
| 37 | \( 1 - 150iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 362T + 6.89e4T^{2} \) |
| 43 | \( 1 - 484iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 280iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 670iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 696T + 2.05e5T^{2} \) |
| 61 | \( 1 - 222T + 2.26e5T^{2} \) |
| 67 | \( 1 + 4iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 96T + 3.57e5T^{2} \) |
| 73 | \( 1 - 178iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 632T + 4.93e5T^{2} \) |
| 83 | \( 1 - 612iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 994T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.63e3iT - 9.12e5T^{2} \) |
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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
−9.609066596854010172545528747345, −8.533539371017494799688240676265, −7.953878536801475725799277886176, −7.03325660006936345752057073369, −5.85056114457770477083531204156, −5.14926164563545981143445552410, −4.40758048122428180016047963412, −3.03503269413336474233032630079, −2.43971861604042363296237140971, −0.60206523867416818480962169817,
0.61428967038398093731425600990, 1.97112195424329039407859023673, 2.90721941915083906025188167616, 4.02672595316898702116303425886, 5.38922975006984331840333063816, 5.70344186151030391049855193375, 7.14622652905843389522028824439, 7.53426842894149902147168880608, 8.349193388379483157296106515950, 9.255258350569777277372392849833
