| L(s) = 1 | + 2-s + 4-s + 4·5-s + 8-s + 4·10-s + 11-s − 13-s + 16-s − 2·17-s + 6·19-s + 4·20-s + 22-s + 2·23-s + 11·25-s − 26-s − 29-s + 4·31-s + 32-s − 2·34-s − 2·37-s + 6·38-s + 4·40-s + 2·41-s + 4·43-s + 44-s + 2·46-s − 2·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 1.78·5-s + 0.353·8-s + 1.26·10-s + 0.301·11-s − 0.277·13-s + 1/4·16-s − 0.485·17-s + 1.37·19-s + 0.894·20-s + 0.213·22-s + 0.417·23-s + 11/5·25-s − 0.196·26-s − 0.185·29-s + 0.718·31-s + 0.176·32-s − 0.342·34-s − 0.328·37-s + 0.973·38-s + 0.632·40-s + 0.312·41-s + 0.609·43-s + 0.150·44-s + 0.294·46-s − 0.291·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.491123637\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.491123637\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 5 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 + 9 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 15 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 5 T + p T^{2} \) |
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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
−7.31438392675144960417326933672, −6.94159338922432209893215457582, −5.98778258620483789705340806073, −5.79674946878444840320552054133, −4.97723577877228460332244783084, −4.41374505793671266183476362534, −3.24346404260257311792450571441, −2.66168009423625437876654531992, −1.85040902287260193872257591100, −1.07528892480629080157626850618,
1.07528892480629080157626850618, 1.85040902287260193872257591100, 2.66168009423625437876654531992, 3.24346404260257311792450571441, 4.41374505793671266183476362534, 4.97723577877228460332244783084, 5.79674946878444840320552054133, 5.98778258620483789705340806073, 6.94159338922432209893215457582, 7.31438392675144960417326933672
