| L(s) = 1 | − 5·11-s − 2·13-s − 6·17-s − 2·19-s − 5·23-s + 5·29-s − 4·31-s − 37-s − 12·41-s + 5·43-s + 2·47-s + 14·53-s − 2·59-s − 5·67-s − 9·71-s + 10·73-s − 11·79-s + 16·83-s − 14·89-s + 8·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 1.50·11-s − 0.554·13-s − 1.45·17-s − 0.458·19-s − 1.04·23-s + 0.928·29-s − 0.718·31-s − 0.164·37-s − 1.87·41-s + 0.762·43-s + 0.291·47-s + 1.92·53-s − 0.260·59-s − 0.610·67-s − 1.06·71-s + 1.17·73-s − 1.23·79-s + 1.75·83-s − 1.48·89-s + 0.812·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 176400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−13.45545908782014, −13.00614622216457, −12.49208570243601, −12.02518622675696, −11.59054048737998, −10.95845065677729, −10.48529811701273, −10.24663436820740, −9.752426498366595, −8.983011037802627, −8.636794651538714, −8.224481099712229, −7.554423866013081, −7.253547642643272, −6.603160782045554, −6.154480348647979, −5.471697932129936, −5.096667333498596, −4.491187348375942, −4.081173997605281, −3.314855991888708, −2.651908067110977, −2.222297406361186, −1.747297693727933, −0.5784839426771572, 0,
0.5784839426771572, 1.747297693727933, 2.222297406361186, 2.651908067110977, 3.314855991888708, 4.081173997605281, 4.491187348375942, 5.096667333498596, 5.471697932129936, 6.154480348647979, 6.603160782045554, 7.253547642643272, 7.554423866013081, 8.224481099712229, 8.636794651538714, 8.983011037802627, 9.752426498366595, 10.24663436820740, 10.48529811701273, 10.95845065677729, 11.59054048737998, 12.02518622675696, 12.49208570243601, 13.00614622216457, 13.45545908782014
