| L(s) = 1 | + 1.73i·3-s − 2.99·9-s − 7·13-s + 5.19i·19-s + 5·25-s − 5.19i·27-s − 8.66i·31-s − 37-s − 12.1i·39-s − 12.1i·43-s − 9·57-s − 14·61-s − 12.1i·67-s − 7·73-s + 8.66i·75-s + ⋯ |
| L(s) = 1 | + 0.999i·3-s − 0.999·9-s − 1.94·13-s + 1.19i·19-s + 25-s − 0.999i·27-s − 1.55i·31-s − 0.164·37-s − 1.94i·39-s − 1.84i·43-s − 1.19·57-s − 1.79·61-s − 1.48i·67-s − 0.819·73-s + 0.999i·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3978417265\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3978417265\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 1.73iT \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 5T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 7T + 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 - 5.19iT - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + 8.66iT - 31T^{2} \) |
| 37 | \( 1 + T + 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + 12.1iT - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 14T + 61T^{2} \) |
| 67 | \( 1 + 12.1iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 7T + 73T^{2} \) |
| 79 | \( 1 + 12.1iT - 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 - 14T + 97T^{2} \) |
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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
−9.053035966969722108950545627598, −8.000943590282717289561775407830, −7.41954774703763249464139903326, −6.33533873425176335963614904931, −5.45255514103565750986780914797, −4.80144729616672893954543188408, −4.01026648894047029139493730596, −3.03115392900874834067879813453, −2.10559207402222168616955538853, −0.13523685308600440766328821257,
1.24460249660826580575757902951, 2.51263898296052585980890995263, 3.03048460560840127825364430557, 4.65571740471496100887281170939, 5.15218771369243878379535029525, 6.25360107049326898120551386870, 7.07186959440517697353335820752, 7.38269028665345657070837334659, 8.366364291848424234648865922414, 9.057388009487985982741926637639
