| L(s) = 1 | + 2-s + 3-s + 4-s + 2·5-s + 6-s − 4·7-s + 8-s + 9-s + 2·10-s − 6·11-s + 12-s − 4·14-s + 2·15-s + 16-s + 17-s + 18-s + 2·19-s + 2·20-s − 4·21-s − 6·22-s + 24-s − 25-s + 27-s − 4·28-s + 2·30-s + 32-s − 6·33-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.894·5-s + 0.408·6-s − 1.51·7-s + 0.353·8-s + 1/3·9-s + 0.632·10-s − 1.80·11-s + 0.288·12-s − 1.06·14-s + 0.516·15-s + 1/4·16-s + 0.242·17-s + 0.235·18-s + 0.458·19-s + 0.447·20-s − 0.872·21-s − 1.27·22-s + 0.204·24-s − 1/5·25-s + 0.192·27-s − 0.755·28-s + 0.365·30-s + 0.176·32-s − 1.04·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 85782 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85782 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 29 | \( 1 \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 9 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 5 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−14.01616392888024, −13.67019544293103, −13.10530409993953, −12.88201518195621, −12.53153933336196, −11.91758109143411, −11.07194231850936, −10.61023688338995, −10.08017954532505, −9.821017293878421, −9.261477379586483, −8.752930971860513, −7.863088351997129, −7.624860472296413, −6.968916705438997, −6.339374072598722, −5.947293096318233, −5.339085299974466, −4.997962167924700, −4.092738473584358, −3.441726438154506, −3.043536659778477, −2.401607206632748, −2.103654145200692, −0.9792725397622926, 0,
0.9792725397622926, 2.103654145200692, 2.401607206632748, 3.043536659778477, 3.441726438154506, 4.092738473584358, 4.997962167924700, 5.339085299974466, 5.947293096318233, 6.339374072598722, 6.968916705438997, 7.624860472296413, 7.863088351997129, 8.752930971860513, 9.261477379586483, 9.821017293878421, 10.08017954532505, 10.61023688338995, 11.07194231850936, 11.91758109143411, 12.53153933336196, 12.88201518195621, 13.10530409993953, 13.67019544293103, 14.01616392888024
