| L(s) = 1 | − 3-s + 9-s − 2·11-s − 2·13-s + 2·17-s − 4·19-s + 8·23-s − 27-s + 4·29-s − 3·31-s + 2·33-s − 9·37-s + 2·39-s + 6·41-s + 43-s + 6·47-s − 2·51-s + 2·53-s + 4·57-s + 6·59-s − 61-s − 12·67-s − 8·69-s − 10·71-s + 73-s − 7·79-s + 81-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s − 0.603·11-s − 0.554·13-s + 0.485·17-s − 0.917·19-s + 1.66·23-s − 0.192·27-s + 0.742·29-s − 0.538·31-s + 0.348·33-s − 1.47·37-s + 0.320·39-s + 0.937·41-s + 0.152·43-s + 0.875·47-s − 0.280·51-s + 0.274·53-s + 0.529·57-s + 0.781·59-s − 0.128·61-s − 1.46·67-s − 0.963·69-s − 1.18·71-s + 0.117·73-s − 0.787·79-s + 1/9·81-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 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 + 9 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 - T + p T^{2} \) |
| 79 | \( 1 + 7 T + p T^{2} \) |
| 83 | \( 1 + 18 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 5 T + p T^{2} \) |
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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
−14.59745172417255, −14.19389471322487, −13.40319588588686, −13.01197293189874, −12.57774577241840, −12.07929048477767, −11.58367333174041, −10.92551651403146, −10.50895175053794, −10.20481519266644, −9.484447180004627, −8.815118837955868, −8.548367124618012, −7.636275847475199, −7.223504237174011, −6.846783942084706, −5.980605662268504, −5.643778143931890, −4.908904451239475, −4.594490147233535, −3.803956712533057, −3.031242666821278, −2.497974002799705, −1.654121863602182, −0.8429839927104695, 0,
0.8429839927104695, 1.654121863602182, 2.497974002799705, 3.031242666821278, 3.803956712533057, 4.594490147233535, 4.908904451239475, 5.643778143931890, 5.980605662268504, 6.846783942084706, 7.223504237174011, 7.636275847475199, 8.548367124618012, 8.815118837955868, 9.484447180004627, 10.20481519266644, 10.50895175053794, 10.92551651403146, 11.58367333174041, 12.07929048477767, 12.57774577241840, 13.01197293189874, 13.40319588588686, 14.19389471322487, 14.59745172417255
