| L(s) = 1 | + 3-s + 9-s − 4·11-s − 13-s − 2·17-s − 4·19-s − 8·23-s + 27-s + 6·29-s + 8·31-s − 4·33-s − 6·37-s − 39-s + 2·41-s + 12·43-s + 8·47-s − 7·49-s − 2·51-s + 2·53-s − 4·57-s + 12·59-s − 2·61-s − 12·67-s − 8·69-s + 16·71-s − 10·73-s + 8·79-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s − 1.20·11-s − 0.277·13-s − 0.485·17-s − 0.917·19-s − 1.66·23-s + 0.192·27-s + 1.11·29-s + 1.43·31-s − 0.696·33-s − 0.986·37-s − 0.160·39-s + 0.312·41-s + 1.82·43-s + 1.16·47-s − 49-s − 0.280·51-s + 0.274·53-s − 0.529·57-s + 1.56·59-s − 0.256·61-s − 1.46·67-s − 0.963·69-s + 1.89·71-s − 1.17·73-s + 0.900·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.917819670\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.917819670\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−7.939428236970305679916521566968, −7.33133212186291559292656053100, −6.43524788089026139039377475493, −5.88592241444235658101323561619, −4.86565535063385090482663143888, −4.36054460607134966378609243541, −3.48486855853978324398418219304, −2.46066849465400339563544193095, −2.15821041599584095280752391820, −0.63876466895030982743652561062,
0.63876466895030982743652561062, 2.15821041599584095280752391820, 2.46066849465400339563544193095, 3.48486855853978324398418219304, 4.36054460607134966378609243541, 4.86565535063385090482663143888, 5.88592241444235658101323561619, 6.43524788089026139039377475493, 7.33133212186291559292656053100, 7.939428236970305679916521566968
