| L(s) = 1 | − 3-s − 7-s + 9-s + 4·11-s − 3·13-s − 4·17-s + 19-s + 21-s − 27-s − 8·29-s − 31-s − 4·33-s + 2·37-s + 3·39-s + 2·41-s + 11·43-s + 2·47-s − 6·49-s + 4·51-s − 10·53-s − 57-s + 6·59-s + 11·61-s − 63-s − 9·67-s − 6·71-s − 14·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.377·7-s + 1/3·9-s + 1.20·11-s − 0.832·13-s − 0.970·17-s + 0.229·19-s + 0.218·21-s − 0.192·27-s − 1.48·29-s − 0.179·31-s − 0.696·33-s + 0.328·37-s + 0.480·39-s + 0.312·41-s + 1.67·43-s + 0.291·47-s − 6/7·49-s + 0.560·51-s − 1.37·53-s − 0.132·57-s + 0.781·59-s + 1.40·61-s − 0.125·63-s − 1.09·67-s − 0.712·71-s − 1.63·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 11 T + p T^{2} \) |
| 67 | \( 1 + 9 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 11 T + p T^{2} \) |
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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
−8.784368778762004334134630846558, −7.57841104555403865715627474095, −7.01388008140161566163525535428, −6.24091338701350002056385194832, −5.54468023306658386558448445677, −4.49977111093898139599169217035, −3.87548099545317697800647945960, −2.64656234407101904449522992425, −1.46980637505125989629154010251, 0,
1.46980637505125989629154010251, 2.64656234407101904449522992425, 3.87548099545317697800647945960, 4.49977111093898139599169217035, 5.54468023306658386558448445677, 6.24091338701350002056385194832, 7.01388008140161566163525535428, 7.57841104555403865715627474095, 8.784368778762004334134630846558
