| L(s) = 1 | − 3-s + 9-s − 6·11-s − 13-s + 3·17-s − 4·19-s − 3·23-s − 27-s + 3·29-s + 5·31-s + 6·33-s + 10·37-s + 39-s − 9·41-s − 43-s − 3·51-s − 9·53-s + 4·57-s + 9·59-s − 11·61-s − 4·67-s + 3·69-s + 12·71-s − 10·73-s + 10·79-s + 81-s − 9·83-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s − 1.80·11-s − 0.277·13-s + 0.727·17-s − 0.917·19-s − 0.625·23-s − 0.192·27-s + 0.557·29-s + 0.898·31-s + 1.04·33-s + 1.64·37-s + 0.160·39-s − 1.40·41-s − 0.152·43-s − 0.420·51-s − 1.23·53-s + 0.529·57-s + 1.17·59-s − 1.40·61-s − 0.488·67-s + 0.361·69-s + 1.42·71-s − 1.17·73-s + 1.12·79-s + 1/9·81-s − 0.987·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 + 11 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−14.73002986022611, −13.96614208805759, −13.52564046452589, −12.98455798215748, −12.58105180403185, −12.12726465039134, −11.49959582172637, −11.03908445910568, −10.40988829594765, −10.05194826041469, −9.782560394805546, −8.784761774256016, −8.320122262185663, −7.650468643827094, −7.551878106521224, −6.472145799766786, −6.223492727452183, −5.551840393215168, −4.872167163134722, −4.679135654625971, −3.787783994139921, −3.001051566345191, −2.477158553851844, −1.755004674222598, −0.7469061219077777, 0,
0.7469061219077777, 1.755004674222598, 2.477158553851844, 3.001051566345191, 3.787783994139921, 4.679135654625971, 4.872167163134722, 5.551840393215168, 6.223492727452183, 6.472145799766786, 7.551878106521224, 7.650468643827094, 8.320122262185663, 8.784761774256016, 9.782560394805546, 10.05194826041469, 10.40988829594765, 11.03908445910568, 11.49959582172637, 12.12726465039134, 12.58105180403185, 12.98455798215748, 13.52564046452589, 13.96614208805759, 14.73002986022611
