| L(s) = 1 | − 3-s + 5-s + 4·7-s + 9-s + 2·13-s − 15-s − 4·21-s + 4·23-s + 25-s − 27-s − 2·29-s + 8·31-s + 4·35-s + 6·37-s − 2·39-s + 6·41-s − 12·43-s + 45-s − 12·47-s + 9·49-s + 10·53-s − 8·59-s − 10·61-s + 4·63-s + 2·65-s + 12·67-s − 4·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1.51·7-s + 1/3·9-s + 0.554·13-s − 0.258·15-s − 0.872·21-s + 0.834·23-s + 1/5·25-s − 0.192·27-s − 0.371·29-s + 1.43·31-s + 0.676·35-s + 0.986·37-s − 0.320·39-s + 0.937·41-s − 1.82·43-s + 0.149·45-s − 1.75·47-s + 9/7·49-s + 1.37·53-s − 1.04·59-s − 1.28·61-s + 0.503·63-s + 0.248·65-s + 1.46·67-s − 0.481·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 277440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 277440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−12.94411892071974, −12.62969222151505, −11.77918675513165, −11.53235970357077, −11.34492098714232, −10.71666883738726, −10.33030496136308, −9.857797228493638, −9.313595636949764, −8.760685354822805, −8.209701685205047, −8.054001569312975, −7.297648068815739, −6.883493543651356, −6.335690332959273, −5.771950402515618, −5.448817394909686, −4.762595126325908, −4.541939298936153, −4.018456699121291, −3.096822247610699, −2.713050567061409, −1.836159859289111, −1.408664699981543, −0.9996516073455753, 0,
0.9996516073455753, 1.408664699981543, 1.836159859289111, 2.713050567061409, 3.096822247610699, 4.018456699121291, 4.541939298936153, 4.762595126325908, 5.448817394909686, 5.771950402515618, 6.335690332959273, 6.883493543651356, 7.297648068815739, 8.054001569312975, 8.209701685205047, 8.760685354822805, 9.313595636949764, 9.857797228493638, 10.33030496136308, 10.71666883738726, 11.34492098714232, 11.53235970357077, 11.77918675513165, 12.62969222151505, 12.94411892071974
