| L(s) = 1 | + 3-s + 2·7-s + 9-s − 4·11-s − 13-s − 2·17-s − 2·19-s + 2·21-s + 27-s − 6·29-s − 10·31-s − 4·33-s − 10·37-s − 39-s + 8·41-s − 4·43-s + 4·47-s − 3·49-s − 2·51-s + 10·53-s − 2·57-s − 8·59-s − 14·61-s + 2·63-s − 2·67-s + 16·71-s + 10·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.755·7-s + 1/3·9-s − 1.20·11-s − 0.277·13-s − 0.485·17-s − 0.458·19-s + 0.436·21-s + 0.192·27-s − 1.11·29-s − 1.79·31-s − 0.696·33-s − 1.64·37-s − 0.160·39-s + 1.24·41-s − 0.609·43-s + 0.583·47-s − 3/7·49-s − 0.280·51-s + 1.37·53-s − 0.264·57-s − 1.04·59-s − 1.79·61-s + 0.251·63-s − 0.244·67-s + 1.89·71-s + 1.17·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 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
−8.007242089652707299494642405749, −7.55475189997539123676200943441, −6.85607792524228677279435904172, −5.68556607312720044752967696384, −5.11884292421876510830098167220, −4.26937900663124280279842805225, −3.39610928208162456611102744465, −2.36732616538951941347129585944, −1.71945211223746090104912796041, 0,
1.71945211223746090104912796041, 2.36732616538951941347129585944, 3.39610928208162456611102744465, 4.26937900663124280279842805225, 5.11884292421876510830098167220, 5.68556607312720044752967696384, 6.85607792524228677279435904172, 7.55475189997539123676200943441, 8.007242089652707299494642405749
