L(s) = 1 | − 2.42·2-s − 3-s + 3.88·4-s − 0.736·5-s + 2.42·6-s − 2.40·7-s − 4.56·8-s + 9-s + 1.78·10-s − 1.03·11-s − 3.88·12-s − 13-s + 5.83·14-s + 0.736·15-s + 3.30·16-s − 2.49·17-s − 2.42·18-s + 0.457·19-s − 2.85·20-s + 2.40·21-s + 2.50·22-s + 7.34·23-s + 4.56·24-s − 4.45·25-s + 2.42·26-s − 27-s − 9.33·28-s + ⋯ |
L(s) = 1 | − 1.71·2-s − 0.577·3-s + 1.94·4-s − 0.329·5-s + 0.990·6-s − 0.909·7-s − 1.61·8-s + 0.333·9-s + 0.565·10-s − 0.311·11-s − 1.12·12-s − 0.277·13-s + 1.55·14-s + 0.190·15-s + 0.825·16-s − 0.604·17-s − 0.571·18-s + 0.104·19-s − 0.639·20-s + 0.525·21-s + 0.533·22-s + 1.53·23-s + 0.931·24-s − 0.891·25-s + 0.475·26-s − 0.192·27-s − 1.76·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2236977103\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2236977103\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 - T \) |
good | 2 | \( 1 + 2.42T + 2T^{2} \) |
| 5 | \( 1 + 0.736T + 5T^{2} \) |
| 7 | \( 1 + 2.40T + 7T^{2} \) |
| 11 | \( 1 + 1.03T + 11T^{2} \) |
| 17 | \( 1 + 2.49T + 17T^{2} \) |
| 19 | \( 1 - 0.457T + 19T^{2} \) |
| 23 | \( 1 - 7.34T + 23T^{2} \) |
| 29 | \( 1 - 4.43T + 29T^{2} \) |
| 31 | \( 1 + 10.0T + 31T^{2} \) |
| 37 | \( 1 + 7.33T + 37T^{2} \) |
| 41 | \( 1 + 9.29T + 41T^{2} \) |
| 43 | \( 1 - 6.10T + 43T^{2} \) |
| 47 | \( 1 - 7.85T + 47T^{2} \) |
| 53 | \( 1 - 2.17T + 53T^{2} \) |
| 59 | \( 1 + 9.64T + 59T^{2} \) |
| 61 | \( 1 - 7.37T + 61T^{2} \) |
| 67 | \( 1 + 2.66T + 67T^{2} \) |
| 71 | \( 1 + 15.6T + 71T^{2} \) |
| 73 | \( 1 - 2.29T + 73T^{2} \) |
| 79 | \( 1 + 15.5T + 79T^{2} \) |
| 83 | \( 1 - 9.92T + 83T^{2} \) |
| 89 | \( 1 - 17.7T + 89T^{2} \) |
| 97 | \( 1 + 5.44T + 97T^{2} \) |
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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
−8.719917461536242592379863549957, −7.63207851346957339111250635200, −7.17058553053368027905563477567, −6.61244979239279409373683041675, −5.75567547102475977481898285224, −4.81224937408784882365334295382, −3.61004141914140521299015072171, −2.65578948271441796967478650382, −1.58550839576594511881269813466, −0.36050079290433118173383664741,
0.36050079290433118173383664741, 1.58550839576594511881269813466, 2.65578948271441796967478650382, 3.61004141914140521299015072171, 4.81224937408784882365334295382, 5.75567547102475977481898285224, 6.61244979239279409373683041675, 7.17058553053368027905563477567, 7.63207851346957339111250635200, 8.719917461536242592379863549957