| L(s) = 1 | + 3-s − 5-s − 7-s + 9-s − 11-s + 2·13-s − 15-s + 2·17-s + 4·19-s − 21-s + 25-s + 27-s − 6·29-s − 33-s + 35-s − 6·37-s + 2·39-s − 6·41-s − 4·43-s − 45-s + 49-s + 2·51-s + 2·53-s + 55-s + 4·57-s + 4·59-s − 6·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s − 0.301·11-s + 0.554·13-s − 0.258·15-s + 0.485·17-s + 0.917·19-s − 0.218·21-s + 1/5·25-s + 0.192·27-s − 1.11·29-s − 0.174·33-s + 0.169·35-s − 0.986·37-s + 0.320·39-s − 0.937·41-s − 0.609·43-s − 0.149·45-s + 1/7·49-s + 0.280·51-s + 0.274·53-s + 0.134·55-s + 0.529·57-s + 0.520·59-s − 0.768·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 73920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 73920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| good | 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 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 | \( 1 - 10 T + p T^{2} \) |
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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
−14.26446594584038, −13.89844228363737, −13.20225886434031, −13.06147864249819, −12.31467459014642, −11.87294720731301, −11.38727741514412, −10.79133546686191, −10.22983270291884, −9.798524595332848, −9.225455087021730, −8.739328192893023, −8.194367614291314, −7.722774422948597, −7.198847464364657, −6.718254485416510, −6.037521759974167, −5.331948643211118, −4.980213450585109, −4.069822429282147, −3.485677559737638, −3.293455163544395, −2.422143363667732, −1.707334616586438, −0.9501807581197928, 0,
0.9501807581197928, 1.707334616586438, 2.422143363667732, 3.293455163544395, 3.485677559737638, 4.069822429282147, 4.980213450585109, 5.331948643211118, 6.037521759974167, 6.718254485416510, 7.198847464364657, 7.722774422948597, 8.194367614291314, 8.739328192893023, 9.225455087021730, 9.798524595332848, 10.22983270291884, 10.79133546686191, 11.38727741514412, 11.87294720731301, 12.31467459014642, 13.06147864249819, 13.20225886434031, 13.89844228363737, 14.26446594584038
