| L(s) = 1 | + 3-s − 4.24·5-s + 3.30·7-s + 9-s + 1.47·11-s + 0.249·13-s − 4.24·15-s − 5.02·17-s + 2.24·19-s + 3.30·21-s − 6.24·23-s + 13.0·25-s + 27-s + 2.41·29-s − 2.89·31-s + 1.47·33-s − 14.0·35-s − 9.71·37-s + 0.249·39-s + 41-s + 10.8·43-s − 4.24·45-s − 4.08·47-s + 3.94·49-s − 5.02·51-s + 1.43·53-s − 6.24·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.90·5-s + 1.25·7-s + 0.333·9-s + 0.443·11-s + 0.0690·13-s − 1.09·15-s − 1.21·17-s + 0.515·19-s + 0.721·21-s − 1.30·23-s + 2.61·25-s + 0.192·27-s + 0.447·29-s − 0.520·31-s + 0.256·33-s − 2.37·35-s − 1.59·37-s + 0.0398·39-s + 0.156·41-s + 1.65·43-s − 0.633·45-s − 0.596·47-s + 0.563·49-s − 0.704·51-s + 0.197·53-s − 0.842·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7872 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 41 | \( 1 - T \) |
| good | 5 | \( 1 + 4.24T + 5T^{2} \) |
| 7 | \( 1 - 3.30T + 7T^{2} \) |
| 11 | \( 1 - 1.47T + 11T^{2} \) |
| 13 | \( 1 - 0.249T + 13T^{2} \) |
| 17 | \( 1 + 5.02T + 17T^{2} \) |
| 19 | \( 1 - 2.24T + 19T^{2} \) |
| 23 | \( 1 + 6.24T + 23T^{2} \) |
| 29 | \( 1 - 2.41T + 29T^{2} \) |
| 31 | \( 1 + 2.89T + 31T^{2} \) |
| 37 | \( 1 + 9.71T + 37T^{2} \) |
| 43 | \( 1 - 10.8T + 43T^{2} \) |
| 47 | \( 1 + 4.08T + 47T^{2} \) |
| 53 | \( 1 - 1.43T + 53T^{2} \) |
| 59 | \( 1 + 2.61T + 59T^{2} \) |
| 61 | \( 1 - 5.71T + 61T^{2} \) |
| 67 | \( 1 + 15.9T + 67T^{2} \) |
| 71 | \( 1 + 10.5T + 71T^{2} \) |
| 73 | \( 1 - 9.39T + 73T^{2} \) |
| 79 | \( 1 + 0.560T + 79T^{2} \) |
| 83 | \( 1 + 3.80T + 83T^{2} \) |
| 89 | \( 1 - 4.11T + 89T^{2} \) |
| 97 | \( 1 + 7.19T + 97T^{2} \) |
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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
−7.53438465795849103412337845668, −7.20390061275731540417057761746, −6.26603312440607413112330880978, −5.10633674893715868549780237465, −4.43432644251531219964803383240, −4.01223706133689026952574662298, −3.29856455381790890864817169648, −2.24997073876659586340146753075, −1.27408469800289589370310109224, 0,
1.27408469800289589370310109224, 2.24997073876659586340146753075, 3.29856455381790890864817169648, 4.01223706133689026952574662298, 4.43432644251531219964803383240, 5.10633674893715868549780237465, 6.26603312440607413112330880978, 7.20390061275731540417057761746, 7.53438465795849103412337845668
