| L(s) = 1 | + 8.28·3-s − 2.38·5-s − 5.83·7-s + 41.6·9-s − 7.33·11-s − 55.6·13-s − 19.7·15-s − 10.0·17-s − 19·19-s − 48.3·21-s + 9.26·23-s − 119.·25-s + 121.·27-s + 83.9·29-s − 202.·31-s − 60.7·33-s + 13.9·35-s − 95.2·37-s − 460.·39-s − 25.9·41-s − 119.·43-s − 99.3·45-s − 467.·47-s − 308.·49-s − 83.1·51-s + 764.·53-s + 17.4·55-s + ⋯ |
| L(s) = 1 | + 1.59·3-s − 0.213·5-s − 0.315·7-s + 1.54·9-s − 0.201·11-s − 1.18·13-s − 0.340·15-s − 0.143·17-s − 0.229·19-s − 0.502·21-s + 0.0839·23-s − 0.954·25-s + 0.866·27-s + 0.537·29-s − 1.17·31-s − 0.320·33-s + 0.0671·35-s − 0.423·37-s − 1.89·39-s − 0.0990·41-s − 0.424·43-s − 0.329·45-s − 1.45·47-s − 0.900·49-s − 0.228·51-s + 1.98·53-s + 0.0428·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + 19T \) |
| good | 3 | \( 1 - 8.28T + 27T^{2} \) |
| 5 | \( 1 + 2.38T + 125T^{2} \) |
| 7 | \( 1 + 5.83T + 343T^{2} \) |
| 11 | \( 1 + 7.33T + 1.33e3T^{2} \) |
| 13 | \( 1 + 55.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 10.0T + 4.91e3T^{2} \) |
| 23 | \( 1 - 9.26T + 1.21e4T^{2} \) |
| 29 | \( 1 - 83.9T + 2.43e4T^{2} \) |
| 31 | \( 1 + 202.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 95.2T + 5.06e4T^{2} \) |
| 41 | \( 1 + 25.9T + 6.89e4T^{2} \) |
| 43 | \( 1 + 119.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 467.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 764.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 69.1T + 2.05e5T^{2} \) |
| 61 | \( 1 - 398.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 243.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 781.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 711.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 723.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.22e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 653.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.69e3T + 9.12e5T^{2} \) |
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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
−8.851450600285601869749524021807, −8.222593017539610200098285230194, −7.44677254089454261902434337008, −6.81306194971548344607273336390, −5.45058433736054954778175445640, −4.36020149065930134835153467229, −3.48988055304392166717979636607, −2.64406969864720718197018877867, −1.79560450384282837257683599167, 0,
1.79560450384282837257683599167, 2.64406969864720718197018877867, 3.48988055304392166717979636607, 4.36020149065930134835153467229, 5.45058433736054954778175445640, 6.81306194971548344607273336390, 7.44677254089454261902434337008, 8.222593017539610200098285230194, 8.851450600285601869749524021807
