| L(s) = 1 | + 3-s − 3·5-s + 9-s + 11-s + 4·13-s − 3·15-s − 4·17-s − 8·23-s + 4·25-s + 27-s − 7·29-s + 11·31-s + 33-s + 4·37-s + 4·39-s + 4·41-s + 2·43-s − 3·45-s − 2·47-s − 4·51-s − 11·53-s − 3·55-s + 7·59-s − 10·61-s − 12·65-s − 10·67-s − 8·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.34·5-s + 1/3·9-s + 0.301·11-s + 1.10·13-s − 0.774·15-s − 0.970·17-s − 1.66·23-s + 4/5·25-s + 0.192·27-s − 1.29·29-s + 1.97·31-s + 0.174·33-s + 0.657·37-s + 0.640·39-s + 0.624·41-s + 0.304·43-s − 0.447·45-s − 0.291·47-s − 0.560·51-s − 1.51·53-s − 0.404·55-s + 0.911·59-s − 1.28·61-s − 1.48·65-s − 1.22·67-s − 0.963·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 7 T + p T^{2} \) |
| 31 | \( 1 - 11 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 11 T + p T^{2} \) |
| 59 | \( 1 - 7 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 - 11 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−7.984331877365611098322205043752, −7.47550962443046423122867307670, −6.48923992128165791411727457072, −5.96251327588168626685822102926, −4.57469865580799387365402434132, −4.11507132647141149122813982159, −3.50490904927474427804969626521, −2.53078320913080985896661665550, −1.37769040801800048545486958028, 0,
1.37769040801800048545486958028, 2.53078320913080985896661665550, 3.50490904927474427804969626521, 4.11507132647141149122813982159, 4.57469865580799387365402434132, 5.96251327588168626685822102926, 6.48923992128165791411727457072, 7.47550962443046423122867307670, 7.984331877365611098322205043752
