| L(s) = 1 | − 2.74·2-s − 3-s + 5.52·4-s − 1.87·5-s + 2.74·6-s − 7-s − 9.67·8-s + 9-s + 5.14·10-s + 5.77·11-s − 5.52·12-s − 4.10·13-s + 2.74·14-s + 1.87·15-s + 15.5·16-s + 1.41·17-s − 2.74·18-s − 0.738·19-s − 10.3·20-s + 21-s − 15.8·22-s + 2.85·23-s + 9.67·24-s − 1.48·25-s + 11.2·26-s − 27-s − 5.52·28-s + ⋯ |
| L(s) = 1 | − 1.94·2-s − 0.577·3-s + 2.76·4-s − 0.838·5-s + 1.12·6-s − 0.377·7-s − 3.42·8-s + 0.333·9-s + 1.62·10-s + 1.74·11-s − 1.59·12-s − 1.13·13-s + 0.733·14-s + 0.484·15-s + 3.87·16-s + 0.343·17-s − 0.646·18-s − 0.169·19-s − 2.31·20-s + 0.218·21-s − 3.37·22-s + 0.594·23-s + 1.97·24-s − 0.296·25-s + 2.20·26-s − 0.192·27-s − 1.04·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 383 | \( 1 + T \) |
| good | 2 | \( 1 + 2.74T + 2T^{2} \) |
| 5 | \( 1 + 1.87T + 5T^{2} \) |
| 11 | \( 1 - 5.77T + 11T^{2} \) |
| 13 | \( 1 + 4.10T + 13T^{2} \) |
| 17 | \( 1 - 1.41T + 17T^{2} \) |
| 19 | \( 1 + 0.738T + 19T^{2} \) |
| 23 | \( 1 - 2.85T + 23T^{2} \) |
| 29 | \( 1 - 1.57T + 29T^{2} \) |
| 31 | \( 1 + 1.45T + 31T^{2} \) |
| 37 | \( 1 + 0.0432T + 37T^{2} \) |
| 41 | \( 1 - 8.90T + 41T^{2} \) |
| 43 | \( 1 + 1.67T + 43T^{2} \) |
| 47 | \( 1 - 0.294T + 47T^{2} \) |
| 53 | \( 1 + 5.28T + 53T^{2} \) |
| 59 | \( 1 + 9.73T + 59T^{2} \) |
| 61 | \( 1 + 7.08T + 61T^{2} \) |
| 67 | \( 1 + 13.2T + 67T^{2} \) |
| 71 | \( 1 - 12.0T + 71T^{2} \) |
| 73 | \( 1 - 2.87T + 73T^{2} \) |
| 79 | \( 1 - 11.9T + 79T^{2} \) |
| 83 | \( 1 + 2.17T + 83T^{2} \) |
| 89 | \( 1 + 3.79T + 89T^{2} \) |
| 97 | \( 1 + 6.29T + 97T^{2} \) |
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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
−7.50307141806107972285848079532, −7.06911726515468564609692485420, −6.42744436915771800498850724086, −5.86293668449917823752227401877, −4.63575374457812214726624429613, −3.69627056660807551974725430678, −2.85563400210622801014825522873, −1.76943579099228874464568239742, −0.914062470252940709138850328396, 0,
0.914062470252940709138850328396, 1.76943579099228874464568239742, 2.85563400210622801014825522873, 3.69627056660807551974725430678, 4.63575374457812214726624429613, 5.86293668449917823752227401877, 6.42744436915771800498850724086, 7.06911726515468564609692485420, 7.50307141806107972285848079532
