| L(s) = 1 | − 2·3-s + 2·5-s − 7-s + 9-s − 2·11-s − 4·13-s − 4·15-s − 6·17-s + 4·19-s + 2·21-s + 23-s − 25-s + 4·27-s − 6·29-s + 4·31-s + 4·33-s − 2·35-s + 8·37-s + 8·39-s + 6·41-s + 6·43-s + 2·45-s + 8·47-s + 49-s + 12·51-s + 4·53-s − 4·55-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.894·5-s − 0.377·7-s + 1/3·9-s − 0.603·11-s − 1.10·13-s − 1.03·15-s − 1.45·17-s + 0.917·19-s + 0.436·21-s + 0.208·23-s − 1/5·25-s + 0.769·27-s − 1.11·29-s + 0.718·31-s + 0.696·33-s − 0.338·35-s + 1.31·37-s + 1.28·39-s + 0.937·41-s + 0.914·43-s + 0.298·45-s + 1.16·47-s + 1/7·49-s + 1.68·51-s + 0.549·53-s − 0.539·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 6 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−16.97119318737866, −16.42402134916296, −15.87204950816534, −15.28796422559443, −14.60231571332903, −13.88192432366202, −13.37779484974007, −12.79268165325685, −12.32506624596113, −11.50295585158313, −11.17092011439315, −10.48360632462261, −9.847429727401846, −9.444008497353435, −8.747064649309010, −7.746161739579152, −7.132289407714452, −6.517443722032271, −5.737059273233733, −5.511436981254922, −4.722724316940364, −4.013862810732974, −2.614277410579654, −2.372075543050787, −0.9766673864714591, 0,
0.9766673864714591, 2.372075543050787, 2.614277410579654, 4.013862810732974, 4.722724316940364, 5.511436981254922, 5.737059273233733, 6.517443722032271, 7.132289407714452, 7.746161739579152, 8.747064649309010, 9.444008497353435, 9.847429727401846, 10.48360632462261, 11.17092011439315, 11.50295585158313, 12.32506624596113, 12.79268165325685, 13.37779484974007, 13.88192432366202, 14.60231571332903, 15.28796422559443, 15.87204950816534, 16.42402134916296, 16.97119318737866
