| L(s) = 1 | + 0.311·3-s + 1.88·5-s − 3.69·7-s − 2.90·9-s − 0.192·13-s + 0.585·15-s + 5.19·17-s − 19-s − 1.14·21-s − 8.28·23-s − 1.45·25-s − 1.83·27-s + 4.23·29-s + 7.33·31-s − 6.95·35-s + 7.90·37-s − 0.0597·39-s + 10.5·41-s + 7.84·43-s − 5.46·45-s + 4.66·47-s + 6.62·49-s + 1.61·51-s − 10.9·53-s − 0.311·57-s − 2.25·59-s + 1.03·61-s + ⋯ |
| L(s) = 1 | + 0.179·3-s + 0.842·5-s − 1.39·7-s − 0.967·9-s − 0.0533·13-s + 0.151·15-s + 1.25·17-s − 0.229·19-s − 0.250·21-s − 1.72·23-s − 0.290·25-s − 0.353·27-s + 0.786·29-s + 1.31·31-s − 1.17·35-s + 1.29·37-s − 0.00957·39-s + 1.64·41-s + 1.19·43-s − 0.815·45-s + 0.680·47-s + 0.946·49-s + 0.226·51-s − 1.50·53-s − 0.0411·57-s − 0.293·59-s + 0.132·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9196 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 0.311T + 3T^{2} \) |
| 5 | \( 1 - 1.88T + 5T^{2} \) |
| 7 | \( 1 + 3.69T + 7T^{2} \) |
| 13 | \( 1 + 0.192T + 13T^{2} \) |
| 17 | \( 1 - 5.19T + 17T^{2} \) |
| 23 | \( 1 + 8.28T + 23T^{2} \) |
| 29 | \( 1 - 4.23T + 29T^{2} \) |
| 31 | \( 1 - 7.33T + 31T^{2} \) |
| 37 | \( 1 - 7.90T + 37T^{2} \) |
| 41 | \( 1 - 10.5T + 41T^{2} \) |
| 43 | \( 1 - 7.84T + 43T^{2} \) |
| 47 | \( 1 - 4.66T + 47T^{2} \) |
| 53 | \( 1 + 10.9T + 53T^{2} \) |
| 59 | \( 1 + 2.25T + 59T^{2} \) |
| 61 | \( 1 - 1.03T + 61T^{2} \) |
| 67 | \( 1 + 4.32T + 67T^{2} \) |
| 71 | \( 1 + 7.50T + 71T^{2} \) |
| 73 | \( 1 + 3.36T + 73T^{2} \) |
| 79 | \( 1 + 9.31T + 79T^{2} \) |
| 83 | \( 1 + 7.76T + 83T^{2} \) |
| 89 | \( 1 + 18.3T + 89T^{2} \) |
| 97 | \( 1 + 4.46T + 97T^{2} \) |
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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
−7.49083299056325952741378914107, −6.37565871847533788476355802043, −5.92517710412388444531639385131, −5.79134326579324741055589081522, −4.50460049050462081733544545833, −3.75592938079168247751968672272, −2.75535231099777801940245433553, −2.57536004488485515425601853019, −1.20246176864900730769587539617, 0,
1.20246176864900730769587539617, 2.57536004488485515425601853019, 2.75535231099777801940245433553, 3.75592938079168247751968672272, 4.50460049050462081733544545833, 5.79134326579324741055589081522, 5.92517710412388444531639385131, 6.37565871847533788476355802043, 7.49083299056325952741378914107
