| L(s) = 1 | − 3-s + 3.38·5-s + 7-s + 9-s + 4.87·11-s + 13-s − 3.38·15-s + 4·17-s − 3.48·19-s − 21-s − 8.26·23-s + 6.48·25-s − 27-s + 5.48·29-s + 6.26·31-s − 4.87·33-s + 3.38·35-s + 8.77·37-s − 39-s − 0.0995·41-s + 2.26·43-s + 3.38·45-s + 10.1·47-s + 49-s − 4·51-s − 12.2·53-s + 16.5·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.51·5-s + 0.377·7-s + 0.333·9-s + 1.47·11-s + 0.277·13-s − 0.875·15-s + 0.970·17-s − 0.800·19-s − 0.218·21-s − 1.72·23-s + 1.29·25-s − 0.192·27-s + 1.01·29-s + 1.12·31-s − 0.849·33-s + 0.572·35-s + 1.44·37-s − 0.160·39-s − 0.0155·41-s + 0.345·43-s + 0.505·45-s + 1.48·47-s + 0.142·49-s − 0.560·51-s − 1.68·53-s + 2.22·55-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
| \(L(1)\) |
\(\approx\) |
\(2.691960058\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.691960058\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 3.38T + 5T^{2} \) |
| 11 | \( 1 - 4.87T + 11T^{2} \) |
| 17 | \( 1 - 4T + 17T^{2} \) |
| 19 | \( 1 + 3.48T + 19T^{2} \) |
| 23 | \( 1 + 8.26T + 23T^{2} \) |
| 29 | \( 1 - 5.48T + 29T^{2} \) |
| 31 | \( 1 - 6.26T + 31T^{2} \) |
| 37 | \( 1 - 8.77T + 37T^{2} \) |
| 41 | \( 1 + 0.0995T + 41T^{2} \) |
| 43 | \( 1 - 2.26T + 43T^{2} \) |
| 47 | \( 1 - 10.1T + 47T^{2} \) |
| 53 | \( 1 + 12.2T + 53T^{2} \) |
| 59 | \( 1 + 6.87T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 6.77T + 67T^{2} \) |
| 71 | \( 1 + 3.65T + 71T^{2} \) |
| 73 | \( 1 + 11.0T + 73T^{2} \) |
| 79 | \( 1 + 13.0T + 79T^{2} \) |
| 83 | \( 1 - 4.61T + 83T^{2} \) |
| 89 | \( 1 + 13.1T + 89T^{2} \) |
| 97 | \( 1 - 8.26T + 97T^{2} \) |
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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
−8.450033008593901750822143395423, −7.62141844125065512312923049979, −6.51621010915714693964580567746, −6.15441434928589940425843294244, −5.71138320051389812729426097311, −4.61859515453645858616956930859, −4.03741385188496781953878445190, −2.73261098569922536553020544383, −1.74156507536735166527069710038, −1.05833964204142587206255468673,
1.05833964204142587206255468673, 1.74156507536735166527069710038, 2.73261098569922536553020544383, 4.03741385188496781953878445190, 4.61859515453645858616956930859, 5.71138320051389812729426097311, 6.15441434928589940425843294244, 6.51621010915714693964580567746, 7.62141844125065512312923049979, 8.450033008593901750822143395423
