| L(s) = 1 | + 2.42·5-s − 4.70·11-s − 3.42·13-s − 1.70·17-s + 1.28·19-s + 1.12·23-s + 0.861·25-s + 4.70·29-s − 3.42·31-s + 8.54·37-s − 3.71·41-s + 5.54·43-s + 11.8·47-s − 10.2·53-s − 11.3·55-s − 4.12·59-s − 9.24·61-s − 8.28·65-s − 11.1·67-s − 14.9·71-s + 2.13·73-s − 14.5·79-s − 8.42·83-s − 4.12·85-s − 16.0·89-s + 3.10·95-s + 16.2·97-s + ⋯ |
| L(s) = 1 | + 1.08·5-s − 1.41·11-s − 0.948·13-s − 0.413·17-s + 0.294·19-s + 0.234·23-s + 0.172·25-s + 0.873·29-s − 0.614·31-s + 1.40·37-s − 0.580·41-s + 0.845·43-s + 1.72·47-s − 1.40·53-s − 1.53·55-s − 0.536·59-s − 1.18·61-s − 1.02·65-s − 1.35·67-s − 1.77·71-s + 0.250·73-s − 1.63·79-s − 0.924·83-s − 0.447·85-s − 1.70·89-s + 0.318·95-s + 1.64·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 2.42T + 5T^{2} \) |
| 11 | \( 1 + 4.70T + 11T^{2} \) |
| 13 | \( 1 + 3.42T + 13T^{2} \) |
| 17 | \( 1 + 1.70T + 17T^{2} \) |
| 19 | \( 1 - 1.28T + 19T^{2} \) |
| 23 | \( 1 - 1.12T + 23T^{2} \) |
| 29 | \( 1 - 4.70T + 29T^{2} \) |
| 31 | \( 1 + 3.42T + 31T^{2} \) |
| 37 | \( 1 - 8.54T + 37T^{2} \) |
| 41 | \( 1 + 3.71T + 41T^{2} \) |
| 43 | \( 1 - 5.54T + 43T^{2} \) |
| 47 | \( 1 - 11.8T + 47T^{2} \) |
| 53 | \( 1 + 10.2T + 53T^{2} \) |
| 59 | \( 1 + 4.12T + 59T^{2} \) |
| 61 | \( 1 + 9.24T + 61T^{2} \) |
| 67 | \( 1 + 11.1T + 67T^{2} \) |
| 71 | \( 1 + 14.9T + 71T^{2} \) |
| 73 | \( 1 - 2.13T + 73T^{2} \) |
| 79 | \( 1 + 14.5T + 79T^{2} \) |
| 83 | \( 1 + 8.42T + 83T^{2} \) |
| 89 | \( 1 + 16.0T + 89T^{2} \) |
| 97 | \( 1 - 16.2T + 97T^{2} \) |
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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
−7.62745228813986881036240842941, −7.32516021429745462083159740903, −6.19493488319327953721802739545, −5.72810912513948336841552031241, −4.96238606197840698588569645860, −4.33974472449841863927900632222, −2.87075250750224091041316154275, −2.54413535985168282813361707824, −1.48065578480195045887138610000, 0,
1.48065578480195045887138610000, 2.54413535985168282813361707824, 2.87075250750224091041316154275, 4.33974472449841863927900632222, 4.96238606197840698588569645860, 5.72810912513948336841552031241, 6.19493488319327953721802739545, 7.32516021429745462083159740903, 7.62745228813986881036240842941
