L(s) = 1 | − 3-s + 2.82·7-s + 9-s + 0.393·11-s + 13-s − 6.21·17-s − 7.34·19-s − 2.82·21-s − 1.44·23-s − 27-s − 0.828·29-s + 1.65·31-s − 0.393·33-s + 6.78·37-s − 39-s + 3.77·41-s − 2.51·43-s + 3.00·47-s + 1.00·49-s + 6.21·51-s − 9.55·53-s + 7.34·57-s + 0.393·59-s + 11.0·61-s + 2.82·63-s + 15.5·67-s + 1.44·69-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1.06·7-s + 0.333·9-s + 0.118·11-s + 0.277·13-s − 1.50·17-s − 1.68·19-s − 0.617·21-s − 0.301·23-s − 0.192·27-s − 0.153·29-s + 0.297·31-s − 0.0684·33-s + 1.11·37-s − 0.160·39-s + 0.590·41-s − 0.383·43-s + 0.438·47-s + 0.142·49-s + 0.869·51-s − 1.31·53-s + 0.972·57-s + 0.0511·59-s + 1.41·61-s + 0.356·63-s + 1.90·67-s + 0.173·69-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
\(L(1)\) |
\(\approx\) |
\(1.563721168\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.563721168\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.82T + 7T^{2} \) |
| 11 | \( 1 - 0.393T + 11T^{2} \) |
| 17 | \( 1 + 6.21T + 17T^{2} \) |
| 19 | \( 1 + 7.34T + 19T^{2} \) |
| 23 | \( 1 + 1.44T + 23T^{2} \) |
| 29 | \( 1 + 0.828T + 29T^{2} \) |
| 31 | \( 1 - 1.65T + 31T^{2} \) |
| 37 | \( 1 - 6.78T + 37T^{2} \) |
| 41 | \( 1 - 3.77T + 41T^{2} \) |
| 43 | \( 1 + 2.51T + 43T^{2} \) |
| 47 | \( 1 - 3.00T + 47T^{2} \) |
| 53 | \( 1 + 9.55T + 53T^{2} \) |
| 59 | \( 1 - 0.393T + 59T^{2} \) |
| 61 | \( 1 - 11.0T + 61T^{2} \) |
| 67 | \( 1 - 15.5T + 67T^{2} \) |
| 71 | \( 1 - 6.56T + 71T^{2} \) |
| 73 | \( 1 - 13.6T + 73T^{2} \) |
| 79 | \( 1 + 14.9T + 79T^{2} \) |
| 83 | \( 1 - 16.4T + 83T^{2} \) |
| 89 | \( 1 + 9.67T + 89T^{2} \) |
| 97 | \( 1 - 0.0418T + 97T^{2} \) |
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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
−8.075609388210134651689339376637, −6.98797344380815643846969735649, −6.47549763215803061315948780793, −5.84726300718988815668639402380, −4.90216820413656520654561007067, −4.44461136302250517848205069694, −3.79695618634887057297061956725, −2.38475630607198249673502468579, −1.85148404412449192725283813141, −0.63666143068376451521559599950,
0.63666143068376451521559599950, 1.85148404412449192725283813141, 2.38475630607198249673502468579, 3.79695618634887057297061956725, 4.44461136302250517848205069694, 4.90216820413656520654561007067, 5.84726300718988815668639402380, 6.47549763215803061315948780793, 6.98797344380815643846969735649, 8.075609388210134651689339376637