| L(s) = 1 | − 3-s + 3.20·5-s − 1.53·7-s + 9-s + 11-s − 13-s − 3.20·15-s + 2.73·17-s − 7.78·19-s + 1.53·21-s − 6.88·23-s + 5.28·25-s − 27-s + 1.22·29-s − 8.11·31-s − 33-s − 4.90·35-s − 6.41·37-s + 39-s + 3.53·41-s + 3.98·43-s + 3.20·45-s − 2.93·47-s − 4.65·49-s − 2.73·51-s − 10.2·53-s + 3.20·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.43·5-s − 0.578·7-s + 0.333·9-s + 0.301·11-s − 0.277·13-s − 0.827·15-s + 0.663·17-s − 1.78·19-s + 0.333·21-s − 1.43·23-s + 1.05·25-s − 0.192·27-s + 0.228·29-s − 1.45·31-s − 0.174·33-s − 0.829·35-s − 1.05·37-s + 0.160·39-s + 0.551·41-s + 0.607·43-s + 0.478·45-s − 0.428·47-s − 0.665·49-s − 0.383·51-s − 1.40·53-s + 0.432·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 - 3.20T + 5T^{2} \) |
| 7 | \( 1 + 1.53T + 7T^{2} \) |
| 17 | \( 1 - 2.73T + 17T^{2} \) |
| 19 | \( 1 + 7.78T + 19T^{2} \) |
| 23 | \( 1 + 6.88T + 23T^{2} \) |
| 29 | \( 1 - 1.22T + 29T^{2} \) |
| 31 | \( 1 + 8.11T + 31T^{2} \) |
| 37 | \( 1 + 6.41T + 37T^{2} \) |
| 41 | \( 1 - 3.53T + 41T^{2} \) |
| 43 | \( 1 - 3.98T + 43T^{2} \) |
| 47 | \( 1 + 2.93T + 47T^{2} \) |
| 53 | \( 1 + 10.2T + 53T^{2} \) |
| 59 | \( 1 - 12.6T + 59T^{2} \) |
| 61 | \( 1 + 6.30T + 61T^{2} \) |
| 67 | \( 1 + 5.80T + 67T^{2} \) |
| 71 | \( 1 - 10.2T + 71T^{2} \) |
| 73 | \( 1 + 15.4T + 73T^{2} \) |
| 79 | \( 1 + 1.98T + 79T^{2} \) |
| 83 | \( 1 - 5.16T + 83T^{2} \) |
| 89 | \( 1 + 4.11T + 89T^{2} \) |
| 97 | \( 1 - 15.9T + 97T^{2} \) |
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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
−8.311955759461148888904259753814, −7.30305188461777599132855708057, −6.43297984348225946563896030157, −6.06390049897871090883340261965, −5.41194078710301494085217118049, −4.45985091221224234044014520091, −3.51490684433814027877649917678, −2.28589566712016555812923411338, −1.61167031335324166418486349828, 0,
1.61167031335324166418486349828, 2.28589566712016555812923411338, 3.51490684433814027877649917678, 4.45985091221224234044014520091, 5.41194078710301494085217118049, 6.06390049897871090883340261965, 6.43297984348225946563896030157, 7.30305188461777599132855708057, 8.311955759461148888904259753814
