| L(s) = 1 | − 2·2-s + 2·4-s − 2·11-s − 13-s − 4·16-s + 2·17-s + 5·19-s + 4·22-s − 6·23-s + 2·26-s − 10·29-s + 3·31-s + 8·32-s − 4·34-s + 2·37-s − 10·38-s − 8·41-s + 43-s − 4·44-s + 12·46-s + 2·47-s − 2·52-s + 4·53-s + 20·58-s − 10·59-s − 7·61-s − 6·62-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s − 0.603·11-s − 0.277·13-s − 16-s + 0.485·17-s + 1.14·19-s + 0.852·22-s − 1.25·23-s + 0.392·26-s − 1.85·29-s + 0.538·31-s + 1.41·32-s − 0.685·34-s + 0.328·37-s − 1.62·38-s − 1.24·41-s + 0.152·43-s − 0.603·44-s + 1.76·46-s + 0.291·47-s − 0.277·52-s + 0.549·53-s + 2.62·58-s − 1.30·59-s − 0.896·61-s − 0.762·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6121186134\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6121186134\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + p T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 17 T + p T^{2} \) |
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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
−16.81026178195627, −16.11327366051779, −15.51065737920429, −15.08460588741975, −14.10800331702639, −13.73285265925789, −13.09736177377511, −12.23090374741989, −11.79679149794108, −11.05212072792146, −10.56505292338071, −9.845837455759072, −9.584539037005880, −8.931236907172527, −8.091767346536184, −7.763158040266260, −7.268588426627047, −6.441544665681545, −5.613241141659605, −4.997691030802374, −4.050252227536214, −3.190079743937692, −2.236585423427908, −1.520151995161236, −0.4617916392926811,
0.4617916392926811, 1.520151995161236, 2.236585423427908, 3.190079743937692, 4.050252227536214, 4.997691030802374, 5.613241141659605, 6.441544665681545, 7.268588426627047, 7.763158040266260, 8.091767346536184, 8.931236907172527, 9.584539037005880, 9.845837455759072, 10.56505292338071, 11.05212072792146, 11.79679149794108, 12.23090374741989, 13.09736177377511, 13.73285265925789, 14.10800331702639, 15.08460588741975, 15.51065737920429, 16.11327366051779, 16.81026178195627
