| L(s) = 1 | − 3-s + 2·7-s + 9-s − 4·11-s + 6·13-s − 4·17-s + 19-s − 2·21-s − 27-s + 10·29-s + 2·31-s + 4·33-s − 2·37-s − 6·39-s + 8·41-s + 8·43-s − 3·49-s + 4·51-s − 6·53-s − 57-s − 2·59-s − 2·61-s + 2·63-s − 4·67-s + 10·73-s − 8·77-s + 2·79-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.755·7-s + 1/3·9-s − 1.20·11-s + 1.66·13-s − 0.970·17-s + 0.229·19-s − 0.436·21-s − 0.192·27-s + 1.85·29-s + 0.359·31-s + 0.696·33-s − 0.328·37-s − 0.960·39-s + 1.24·41-s + 1.21·43-s − 3/7·49-s + 0.560·51-s − 0.824·53-s − 0.132·57-s − 0.260·59-s − 0.256·61-s + 0.251·63-s − 0.488·67-s + 1.17·73-s − 0.911·77-s + 0.225·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 - 10 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 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
−13.93337541121858, −13.70935884529227, −12.97430877981146, −12.79109971386621, −12.00983595715022, −11.63336697893056, −10.94760769485248, −10.72386722902527, −10.50471369152610, −9.586952511658339, −9.124630923265917, −8.432858535949461, −8.108900712898232, −7.666186078191138, −6.908641602235698, −6.299550366090333, −6.043013066348774, −5.229532151057199, −4.891119854239197, −4.276777248528444, −3.749312266437701, −2.818637144466461, −2.408111980604780, −1.426652768040919, −0.9792364189863016, 0,
0.9792364189863016, 1.426652768040919, 2.408111980604780, 2.818637144466461, 3.749312266437701, 4.276777248528444, 4.891119854239197, 5.229532151057199, 6.043013066348774, 6.299550366090333, 6.908641602235698, 7.666186078191138, 8.108900712898232, 8.432858535949461, 9.124630923265917, 9.586952511658339, 10.50471369152610, 10.72386722902527, 10.94760769485248, 11.63336697893056, 12.00983595715022, 12.79109971386621, 12.97430877981146, 13.70935884529227, 13.93337541121858
