L(s) = 1 | − 3-s + 2·5-s − 4·7-s + 9-s − 4·11-s − 2·13-s − 2·15-s + 2·17-s + 19-s + 4·21-s − 4·23-s − 25-s − 27-s + 6·29-s + 4·33-s − 8·35-s − 2·37-s + 2·39-s − 6·41-s + 12·43-s + 2·45-s − 4·47-s + 9·49-s − 2·51-s − 2·53-s − 8·55-s − 57-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.894·5-s − 1.51·7-s + 1/3·9-s − 1.20·11-s − 0.554·13-s − 0.516·15-s + 0.485·17-s + 0.229·19-s + 0.872·21-s − 0.834·23-s − 1/5·25-s − 0.192·27-s + 1.11·29-s + 0.696·33-s − 1.35·35-s − 0.328·37-s + 0.320·39-s − 0.937·41-s + 1.82·43-s + 0.298·45-s − 0.583·47-s + 9/7·49-s − 0.280·51-s − 0.274·53-s − 1.07·55-s − 0.132·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.033090832\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.033090832\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 19 | \( 1 - T \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−8.575581379081595946399906290030, −7.68315992600211119281045687216, −6.89450560109559882763231283410, −6.22878034446784481943906096825, −5.61348919909799078069556151325, −5.01609967301778459040181934282, −3.84386755180305928698130413093, −2.88341144922637099548718869524, −2.13598639102077654585928979729, −0.58134785906834370168320752722,
0.58134785906834370168320752722, 2.13598639102077654585928979729, 2.88341144922637099548718869524, 3.84386755180305928698130413093, 5.01609967301778459040181934282, 5.61348919909799078069556151325, 6.22878034446784481943906096825, 6.89450560109559882763231283410, 7.68315992600211119281045687216, 8.575581379081595946399906290030