| L(s) = 1 | + 3-s − 0.152·5-s + 7-s + 9-s − 0.385·11-s + 13-s − 0.152·15-s + 7.43·17-s + 7.20·19-s + 21-s − 2.90·23-s − 4.97·25-s + 27-s − 5.20·29-s − 1.76·31-s − 0.385·33-s − 0.152·35-s + 7.43·37-s + 39-s + 7.05·41-s − 2.90·43-s − 0.152·45-s + 3.59·47-s + 49-s + 7.43·51-s − 10.9·53-s + 0.0587·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.0681·5-s + 0.377·7-s + 0.333·9-s − 0.116·11-s + 0.277·13-s − 0.0393·15-s + 1.80·17-s + 1.65·19-s + 0.218·21-s − 0.604·23-s − 0.995·25-s + 0.192·27-s − 0.966·29-s − 0.317·31-s − 0.0670·33-s − 0.0257·35-s + 1.22·37-s + 0.160·39-s + 1.10·41-s − 0.442·43-s − 0.0227·45-s + 0.523·47-s + 0.142·49-s + 1.04·51-s − 1.50·53-s + 0.00791·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.781769020\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.781769020\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 + 0.152T + 5T^{2} \) |
| 11 | \( 1 + 0.385T + 11T^{2} \) |
| 17 | \( 1 - 7.43T + 17T^{2} \) |
| 19 | \( 1 - 7.20T + 19T^{2} \) |
| 23 | \( 1 + 2.90T + 23T^{2} \) |
| 29 | \( 1 + 5.20T + 29T^{2} \) |
| 31 | \( 1 + 1.76T + 31T^{2} \) |
| 37 | \( 1 - 7.43T + 37T^{2} \) |
| 41 | \( 1 - 7.05T + 41T^{2} \) |
| 43 | \( 1 + 2.90T + 43T^{2} \) |
| 47 | \( 1 - 3.59T + 47T^{2} \) |
| 53 | \( 1 + 10.9T + 53T^{2} \) |
| 59 | \( 1 - 5.82T + 59T^{2} \) |
| 61 | \( 1 - 12.5T + 61T^{2} \) |
| 67 | \( 1 + 9.80T + 67T^{2} \) |
| 71 | \( 1 + 5.82T + 71T^{2} \) |
| 73 | \( 1 - 3.09T + 73T^{2} \) |
| 79 | \( 1 + 12.6T + 79T^{2} \) |
| 83 | \( 1 + 11.8T + 83T^{2} \) |
| 89 | \( 1 + 5.59T + 89T^{2} \) |
| 97 | \( 1 - 18.9T + 97T^{2} \) |
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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
−8.218521649699872673495367813419, −7.58197402140931577610717517430, −7.35414213614108625270865306605, −5.90422258242948801682912218401, −5.60139204724817769209709674723, −4.53613160882751697037217117831, −3.64798669564500641208541952217, −3.05824327428536935663769153341, −1.92104806739094324493716845180, −0.967781568455477424527698329537,
0.967781568455477424527698329537, 1.92104806739094324493716845180, 3.05824327428536935663769153341, 3.64798669564500641208541952217, 4.53613160882751697037217117831, 5.60139204724817769209709674723, 5.90422258242948801682912218401, 7.35414213614108625270865306605, 7.58197402140931577610717517430, 8.218521649699872673495367813419
