| L(s) = 1 | − 3-s − 5-s + 0.585·7-s + 9-s − 1.41·11-s − 4.24·13-s + 15-s − 2.82·17-s + 19-s − 0.585·21-s − 0.828·23-s + 25-s − 27-s − 6.24·29-s + 3.17·31-s + 1.41·33-s − 0.585·35-s + 7.07·37-s + 4.24·39-s − 6.24·41-s + 10.2·43-s − 45-s + 0.828·47-s − 6.65·49-s + 2.82·51-s + 4.48·53-s + 1.41·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.221·7-s + 0.333·9-s − 0.426·11-s − 1.17·13-s + 0.258·15-s − 0.685·17-s + 0.229·19-s − 0.127·21-s − 0.172·23-s + 0.200·25-s − 0.192·27-s − 1.15·29-s + 0.569·31-s + 0.246·33-s − 0.0990·35-s + 1.16·37-s + 0.679·39-s − 0.974·41-s + 1.56·43-s − 0.149·45-s + 0.120·47-s − 0.950·49-s + 0.396·51-s + 0.616·53-s + 0.190·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9349768575\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9349768575\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 7 | \( 1 - 0.585T + 7T^{2} \) |
| 11 | \( 1 + 1.41T + 11T^{2} \) |
| 13 | \( 1 + 4.24T + 13T^{2} \) |
| 17 | \( 1 + 2.82T + 17T^{2} \) |
| 23 | \( 1 + 0.828T + 23T^{2} \) |
| 29 | \( 1 + 6.24T + 29T^{2} \) |
| 31 | \( 1 - 3.17T + 31T^{2} \) |
| 37 | \( 1 - 7.07T + 37T^{2} \) |
| 41 | \( 1 + 6.24T + 41T^{2} \) |
| 43 | \( 1 - 10.2T + 43T^{2} \) |
| 47 | \( 1 - 0.828T + 47T^{2} \) |
| 53 | \( 1 - 4.48T + 53T^{2} \) |
| 59 | \( 1 + 2.82T + 59T^{2} \) |
| 61 | \( 1 + 8T + 61T^{2} \) |
| 67 | \( 1 + 11.3T + 67T^{2} \) |
| 71 | \( 1 - 10.8T + 71T^{2} \) |
| 73 | \( 1 - 7.65T + 73T^{2} \) |
| 79 | \( 1 - 13.6T + 79T^{2} \) |
| 83 | \( 1 - 6T + 83T^{2} \) |
| 89 | \( 1 - 17.5T + 89T^{2} \) |
| 97 | \( 1 + 9.89T + 97T^{2} \) |
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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
−8.051770561901685873738901575430, −7.66953936516252925853522840146, −6.91076218640348467049664668994, −6.15199153696383977971176696767, −5.27235523143529734396562364263, −4.69661079306401521636875259251, −3.95329200373950509739396774569, −2.83864412546053812854048244951, −1.94355958571798899954313066482, −0.53727142922983314127949709546,
0.53727142922983314127949709546, 1.94355958571798899954313066482, 2.83864412546053812854048244951, 3.95329200373950509739396774569, 4.69661079306401521636875259251, 5.27235523143529734396562364263, 6.15199153696383977971176696767, 6.91076218640348467049664668994, 7.66953936516252925853522840146, 8.051770561901685873738901575430
