| L(s) = 1 | − 5-s − 2·7-s + 4·11-s − 2·13-s − 6·17-s + 2·19-s + 23-s + 25-s + 4·29-s + 8·31-s + 2·35-s + 2·37-s − 6·43-s + 4·47-s − 3·49-s − 10·53-s − 4·55-s − 4·59-s − 6·61-s + 2·65-s − 10·67-s + 8·71-s − 10·73-s − 8·77-s − 14·79-s + 4·83-s + 6·85-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.755·7-s + 1.20·11-s − 0.554·13-s − 1.45·17-s + 0.458·19-s + 0.208·23-s + 1/5·25-s + 0.742·29-s + 1.43·31-s + 0.338·35-s + 0.328·37-s − 0.914·43-s + 0.583·47-s − 3/7·49-s − 1.37·53-s − 0.539·55-s − 0.520·59-s − 0.768·61-s + 0.248·65-s − 1.22·67-s + 0.949·71-s − 1.17·73-s − 0.911·77-s − 1.57·79-s + 0.439·83-s + 0.650·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−8.090753670292665678650630726104, −7.19318728875240680365809782499, −6.56603709386798265186780548010, −6.12352424703812886192007841034, −4.80136783253141896105575045128, −4.34277649858975340857680013722, −3.35934720941182887444251376190, −2.61752725197143501613702660246, −1.32657258305720663270738938186, 0,
1.32657258305720663270738938186, 2.61752725197143501613702660246, 3.35934720941182887444251376190, 4.34277649858975340857680013722, 4.80136783253141896105575045128, 6.12352424703812886192007841034, 6.56603709386798265186780548010, 7.19318728875240680365809782499, 8.090753670292665678650630726104
