| L(s) = 1 | + i·3-s − 0.0883i·7-s − 9-s + 2.26·11-s − 2.65i·13-s + 2.08i·17-s − 1.76·19-s + 0.0883·21-s − 4.74i·23-s − i·27-s − 3.70·29-s − 4.10·31-s + 2.26i·33-s + 7.11i·37-s + 2.65·39-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s − 0.0333i·7-s − 0.333·9-s + 0.684·11-s − 0.735i·13-s + 0.506i·17-s − 0.403·19-s + 0.0192·21-s − 0.988i·23-s − 0.192i·27-s − 0.688·29-s − 0.737·31-s + 0.395i·33-s + 1.17i·37-s + 0.424·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7500 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.578239408\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.578239408\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 0.0883iT - 7T^{2} \) |
| 11 | \( 1 - 2.26T + 11T^{2} \) |
| 13 | \( 1 + 2.65iT - 13T^{2} \) |
| 17 | \( 1 - 2.08iT - 17T^{2} \) |
| 19 | \( 1 + 1.76T + 19T^{2} \) |
| 23 | \( 1 + 4.74iT - 23T^{2} \) |
| 29 | \( 1 + 3.70T + 29T^{2} \) |
| 31 | \( 1 + 4.10T + 31T^{2} \) |
| 37 | \( 1 - 7.11iT - 37T^{2} \) |
| 41 | \( 1 + 6.58T + 41T^{2} \) |
| 43 | \( 1 + 1.79iT - 43T^{2} \) |
| 47 | \( 1 - 10.1iT - 47T^{2} \) |
| 53 | \( 1 - 0.961iT - 53T^{2} \) |
| 59 | \( 1 - 8.97T + 59T^{2} \) |
| 61 | \( 1 - 9.46T + 61T^{2} \) |
| 67 | \( 1 - 13.9iT - 67T^{2} \) |
| 71 | \( 1 - 6.14T + 71T^{2} \) |
| 73 | \( 1 - 3.15iT - 73T^{2} \) |
| 79 | \( 1 + 13.3T + 79T^{2} \) |
| 83 | \( 1 - 15.8iT - 83T^{2} \) |
| 89 | \( 1 - 10.2T + 89T^{2} \) |
| 97 | \( 1 + 12.8iT - 97T^{2} \) |
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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
−8.375832891136483907928338021775, −7.30442271877207718951114893757, −6.66731130040106847412306472776, −5.88372424589306001419659187480, −5.29505804593268066189092163989, −4.38719634470437495730103144835, −3.84004173058444787819740837047, −3.02324835442847155692374960937, −2.09460586146273093755840535270, −0.928073607746757527282172208761,
0.43805302985775045110015409207, 1.67968609808456176820300197808, 2.21831621276732741959444853555, 3.46013791912542148369245171262, 3.96134138972607099543236674826, 5.01853729517239439999207729017, 5.65585594371694153850752716743, 6.44414257096689056136371251494, 7.09372824158464483236356161434, 7.50152860434563216462277963535
