| L(s) = 1 | + 3-s − 1.41·5-s − 7-s + 9-s − 2·11-s − 2.82·13-s − 1.41·15-s + 4.24·17-s − 19-s − 21-s + 3.65·23-s − 2.99·25-s + 27-s + 0.585·29-s + 4.82·31-s − 2·33-s + 1.41·35-s + 10.4·37-s − 2.82·39-s − 1.65·41-s − 2·43-s − 1.41·45-s − 7.07·47-s + 49-s + 4.24·51-s + 11.8·53-s + 2.82·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.632·5-s − 0.377·7-s + 0.333·9-s − 0.603·11-s − 0.784·13-s − 0.365·15-s + 1.02·17-s − 0.229·19-s − 0.218·21-s + 0.762·23-s − 0.599·25-s + 0.192·27-s + 0.108·29-s + 0.867·31-s − 0.348·33-s + 0.239·35-s + 1.72·37-s − 0.452·39-s − 0.258·41-s − 0.304·43-s − 0.210·45-s − 1.03·47-s + 0.142·49-s + 0.594·51-s + 1.63·53-s + 0.381·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 + 1.41T + 5T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 + 2.82T + 13T^{2} \) |
| 17 | \( 1 - 4.24T + 17T^{2} \) |
| 23 | \( 1 - 3.65T + 23T^{2} \) |
| 29 | \( 1 - 0.585T + 29T^{2} \) |
| 31 | \( 1 - 4.82T + 31T^{2} \) |
| 37 | \( 1 - 10.4T + 37T^{2} \) |
| 41 | \( 1 + 1.65T + 41T^{2} \) |
| 43 | \( 1 + 2T + 43T^{2} \) |
| 47 | \( 1 + 7.07T + 47T^{2} \) |
| 53 | \( 1 - 11.8T + 53T^{2} \) |
| 59 | \( 1 + 13.6T + 59T^{2} \) |
| 61 | \( 1 - 0.828T + 61T^{2} \) |
| 67 | \( 1 - 4.48T + 67T^{2} \) |
| 71 | \( 1 + 15.4T + 71T^{2} \) |
| 73 | \( 1 + 11.6T + 73T^{2} \) |
| 79 | \( 1 + 9.17T + 79T^{2} \) |
| 83 | \( 1 + 1.41T + 83T^{2} \) |
| 89 | \( 1 + 4T + 89T^{2} \) |
| 97 | \( 1 + 3.65T + 97T^{2} \) |
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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
−7.69012558883546554116202852320, −7.22760572315710706043676485458, −6.32658024128316138808338499600, −5.50239400413732663409306791771, −4.66407785287026059946551961164, −3.99969159642597611147208598920, −3.03497585082454342142071083716, −2.61454803164769045471429815936, −1.29859444877551390333051712875, 0,
1.29859444877551390333051712875, 2.61454803164769045471429815936, 3.03497585082454342142071083716, 3.99969159642597611147208598920, 4.66407785287026059946551961164, 5.50239400413732663409306791771, 6.32658024128316138808338499600, 7.22760572315710706043676485458, 7.69012558883546554116202852320
