| L(s) = 1 | − 2.44·3-s − 3.46·5-s − 2.82·7-s + 2.99·9-s − 2.44·11-s + 3.46·13-s + 8.48·15-s + 4·17-s + 2.44·19-s + 6.92·21-s − 2.82·23-s + 6.99·25-s − 3.46·29-s − 5.65·31-s + 5.99·33-s + 9.79·35-s + 3.46·37-s − 8.48·39-s − 2·41-s + 12.2·43-s − 10.3·45-s + 11.3·47-s + 1.00·49-s − 9.79·51-s − 10.3·53-s + 8.48·55-s − 5.99·57-s + ⋯ |
| L(s) = 1 | − 1.41·3-s − 1.54·5-s − 1.06·7-s + 0.999·9-s − 0.738·11-s + 0.960·13-s + 2.19·15-s + 0.970·17-s + 0.561·19-s + 1.51·21-s − 0.589·23-s + 1.39·25-s − 0.643·29-s − 1.01·31-s + 1.04·33-s + 1.65·35-s + 0.569·37-s − 1.35·39-s − 0.312·41-s + 1.86·43-s − 1.54·45-s + 1.65·47-s + 0.142·49-s − 1.37·51-s − 1.42·53-s + 1.14·55-s − 0.794·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4170613586\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4170613586\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 + 2.44T + 3T^{2} \) |
| 5 | \( 1 + 3.46T + 5T^{2} \) |
| 7 | \( 1 + 2.82T + 7T^{2} \) |
| 11 | \( 1 + 2.44T + 11T^{2} \) |
| 13 | \( 1 - 3.46T + 13T^{2} \) |
| 17 | \( 1 - 4T + 17T^{2} \) |
| 19 | \( 1 - 2.44T + 19T^{2} \) |
| 23 | \( 1 + 2.82T + 23T^{2} \) |
| 29 | \( 1 + 3.46T + 29T^{2} \) |
| 31 | \( 1 + 5.65T + 31T^{2} \) |
| 37 | \( 1 - 3.46T + 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 - 12.2T + 43T^{2} \) |
| 47 | \( 1 - 11.3T + 47T^{2} \) |
| 53 | \( 1 + 10.3T + 53T^{2} \) |
| 59 | \( 1 - 2.44T + 59T^{2} \) |
| 61 | \( 1 - 3.46T + 61T^{2} \) |
| 67 | \( 1 + 7.34T + 67T^{2} \) |
| 71 | \( 1 - 2.82T + 71T^{2} \) |
| 73 | \( 1 - 8T + 73T^{2} \) |
| 79 | \( 1 + 5.65T + 79T^{2} \) |
| 83 | \( 1 - 7.34T + 83T^{2} \) |
| 89 | \( 1 + 8T + 89T^{2} \) |
| 97 | \( 1 - 12T + 97T^{2} \) |
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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
−11.02786644603350626731985498827, −10.35506766564006734300632735802, −9.213602825676252923991223225768, −7.925601852945032905817741070741, −7.26817370569359894767079006764, −6.14599753329153674937562590492, −5.43714576545236826468004254293, −4.14116519744998001430897462523, −3.26483524293334446935560547667, −0.60788101583098772276831992838,
0.60788101583098772276831992838, 3.26483524293334446935560547667, 4.14116519744998001430897462523, 5.43714576545236826468004254293, 6.14599753329153674937562590492, 7.26817370569359894767079006764, 7.925601852945032905817741070741, 9.213602825676252923991223225768, 10.35506766564006734300632735802, 11.02786644603350626731985498827
