| L(s) = 1 | − 1.41·3-s + 2.82·5-s − 7-s − 0.999·9-s − 2.82·11-s − 4.00·15-s + 2·17-s − 4.24·19-s + 1.41·21-s − 4·23-s + 3.00·25-s + 5.65·27-s + 9.89·29-s + 6·31-s + 4.00·33-s − 2.82·35-s − 4.24·37-s + 2·41-s − 8.48·43-s − 2.82·45-s + 10·47-s + 49-s − 2.82·51-s + 7.07·53-s − 8.00·55-s + 6·57-s + 9.89·59-s + ⋯ |
| L(s) = 1 | − 0.816·3-s + 1.26·5-s − 0.377·7-s − 0.333·9-s − 0.852·11-s − 1.03·15-s + 0.485·17-s − 0.973·19-s + 0.308·21-s − 0.834·23-s + 0.600·25-s + 1.08·27-s + 1.83·29-s + 1.07·31-s + 0.696·33-s − 0.478·35-s − 0.697·37-s + 0.312·41-s − 1.29·43-s − 0.421·45-s + 1.45·47-s + 0.142·49-s − 0.396·51-s + 0.971·53-s − 1.07·55-s + 0.794·57-s + 1.28·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 3 | \( 1 + 1.41T + 3T^{2} \) |
| 5 | \( 1 - 2.82T + 5T^{2} \) |
| 11 | \( 1 + 2.82T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 + 4.24T + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 - 9.89T + 29T^{2} \) |
| 31 | \( 1 - 6T + 31T^{2} \) |
| 37 | \( 1 + 4.24T + 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 + 8.48T + 43T^{2} \) |
| 47 | \( 1 - 10T + 47T^{2} \) |
| 53 | \( 1 - 7.07T + 53T^{2} \) |
| 59 | \( 1 - 9.89T + 59T^{2} \) |
| 61 | \( 1 - 2.82T + 61T^{2} \) |
| 67 | \( 1 + 11.3T + 67T^{2} \) |
| 71 | \( 1 + 4T + 71T^{2} \) |
| 73 | \( 1 + 14T + 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 - 4.24T + 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 + 6T + 97T^{2} \) |
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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
−7.47056909924355517447061245876, −6.52138232495428434841982906046, −6.19054009837139446134624406396, −5.54248329119735670525248323009, −5.01027775653536162656469352537, −4.11507023733075879084176989903, −2.84872204456887234487792132295, −2.39016782711388368491191976576, −1.19690785896610356580151388728, 0,
1.19690785896610356580151388728, 2.39016782711388368491191976576, 2.84872204456887234487792132295, 4.11507023733075879084176989903, 5.01027775653536162656469352537, 5.54248329119735670525248323009, 6.19054009837139446134624406396, 6.52138232495428434841982906046, 7.47056909924355517447061245876
