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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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
−7.70863690197081641300923315867, −7.43768377045690539371064192149, −6.32659243271786982833107244701, −5.73511700927854591084284829550, −4.87671606017880943818782848404, −4.52168645858656286596220017915, −3.19979097335244710364858827163, −2.45520929275462982861139256959, −1.26563718094136118429355723760, 0,
1.26563718094136118429355723760, 2.45520929275462982861139256959, 3.19979097335244710364858827163, 4.52168645858656286596220017915, 4.87671606017880943818782848404, 5.73511700927854591084284829550, 6.32659243271786982833107244701, 7.43768377045690539371064192149, 7.70863690197081641300923315867