| L(s) = 1 | + 3·3-s − 5·5-s + 9·9-s − 15·15-s − 14·17-s + 22·19-s + 34·23-s + 25·25-s + 27·27-s + 2·31-s − 45·45-s − 14·47-s + 49·49-s − 42·51-s + 86·53-s + 66·57-s + 118·61-s + 102·69-s + 75·75-s + 98·79-s + 81·81-s − 154·83-s + 70·85-s + 6·93-s − 110·95-s − 106·107-s + 22·109-s + ⋯ |
| L(s) = 1 | + 3-s − 5-s + 9-s − 15-s − 0.823·17-s + 1.15·19-s + 1.47·23-s + 25-s + 27-s + 2/31·31-s − 45-s − 0.297·47-s + 49-s − 0.823·51-s + 1.62·53-s + 1.15·57-s + 1.93·61-s + 1.47·69-s + 75-s + 1.24·79-s + 81-s − 1.85·83-s + 0.823·85-s + 2/31·93-s − 1.15·95-s − 0.990·107-s + 0.201·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.406862285\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.406862285\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p T \) |
| 5 | \( 1 + p T \) |
| good | 7 | \( ( 1 - p T )( 1 + p T ) \) |
| 11 | \( ( 1 - p T )( 1 + p T ) \) |
| 13 | \( ( 1 - p T )( 1 + p T ) \) |
| 17 | \( 1 + 14 T + p^{2} T^{2} \) |
| 19 | \( 1 - 22 T + p^{2} T^{2} \) |
| 23 | \( 1 - 34 T + p^{2} T^{2} \) |
| 29 | \( ( 1 - p T )( 1 + p T ) \) |
| 31 | \( 1 - 2 T + p^{2} T^{2} \) |
| 37 | \( ( 1 - p T )( 1 + p T ) \) |
| 41 | \( ( 1 - p T )( 1 + p T ) \) |
| 43 | \( ( 1 - p T )( 1 + p T ) \) |
| 47 | \( 1 + 14 T + p^{2} T^{2} \) |
| 53 | \( 1 - 86 T + p^{2} T^{2} \) |
| 59 | \( ( 1 - p T )( 1 + p T ) \) |
| 61 | \( 1 - 118 T + p^{2} T^{2} \) |
| 67 | \( ( 1 - p T )( 1 + p T ) \) |
| 71 | \( ( 1 - p T )( 1 + p T ) \) |
| 73 | \( ( 1 - p T )( 1 + p T ) \) |
| 79 | \( 1 - 98 T + p^{2} T^{2} \) |
| 83 | \( 1 + 154 T + p^{2} T^{2} \) |
| 89 | \( ( 1 - p T )( 1 + p T ) \) |
| 97 | \( ( 1 - p T )( 1 + p T ) \) |
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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
−9.650656002337281562464649995460, −8.856335304749391979132766625134, −8.268705780996585607452380261920, −7.28880001385535669069123991695, −6.88117166867882854200098686257, −5.26939005940498076423117612453, −4.30022219795605510970309119756, −3.44770007742856349128488020872, −2.53297145717665903502010451478, −0.964153572142294797891098416898,
0.964153572142294797891098416898, 2.53297145717665903502010451478, 3.44770007742856349128488020872, 4.30022219795605510970309119756, 5.26939005940498076423117612453, 6.88117166867882854200098686257, 7.28880001385535669069123991695, 8.268705780996585607452380261920, 8.856335304749391979132766625134, 9.650656002337281562464649995460
