| L(s) = 1 | − 3-s − 4·7-s + 9-s + 4·11-s + 2·13-s + 4·19-s + 4·21-s − 5·25-s − 27-s + 6·29-s − 8·31-s − 4·33-s + 6·37-s − 2·39-s − 2·43-s − 8·47-s + 9·49-s + 8·53-s − 4·57-s + 10·61-s − 4·63-s − 14·67-s + 12·71-s − 2·73-s + 5·75-s − 16·77-s − 2·79-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.51·7-s + 1/3·9-s + 1.20·11-s + 0.554·13-s + 0.917·19-s + 0.872·21-s − 25-s − 0.192·27-s + 1.11·29-s − 1.43·31-s − 0.696·33-s + 0.986·37-s − 0.320·39-s − 0.304·43-s − 1.16·47-s + 9/7·49-s + 1.09·53-s − 0.529·57-s + 1.28·61-s − 0.503·63-s − 1.71·67-s + 1.42·71-s − 0.234·73-s + 0.577·75-s − 1.82·77-s − 0.225·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.309198561\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.309198561\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 167 | \( 1 - T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−7.68996557558252642783672355318, −6.89967187721694149878700621787, −6.45855797550976293851250943366, −5.90549662075292049802460164544, −5.19636634665212503382252581218, −4.06587320244688895933008338686, −3.65813846497706608760980555419, −2.82245760088355938139776327953, −1.58297242578836055475646597815, −0.60389804136970123600456008522,
0.60389804136970123600456008522, 1.58297242578836055475646597815, 2.82245760088355938139776327953, 3.65813846497706608760980555419, 4.06587320244688895933008338686, 5.19636634665212503382252581218, 5.90549662075292049802460164544, 6.45855797550976293851250943366, 6.89967187721694149878700621787, 7.68996557558252642783672355318
