L(s) = 1 | + 1.93·2-s + 3-s + 1.74·4-s − 3.45·5-s + 1.93·6-s − 7-s − 0.498·8-s + 9-s − 6.67·10-s + 1.25·11-s + 1.74·12-s − 4.36·13-s − 1.93·14-s − 3.45·15-s − 4.44·16-s + 0.181·17-s + 1.93·18-s + 0.648·19-s − 6.01·20-s − 21-s + 2.42·22-s + 2.90·23-s − 0.498·24-s + 6.91·25-s − 8.44·26-s + 27-s − 1.74·28-s + ⋯ |
L(s) = 1 | + 1.36·2-s + 0.577·3-s + 0.871·4-s − 1.54·5-s + 0.789·6-s − 0.377·7-s − 0.176·8-s + 0.333·9-s − 2.11·10-s + 0.377·11-s + 0.502·12-s − 1.21·13-s − 0.516·14-s − 0.891·15-s − 1.11·16-s + 0.0440·17-s + 0.455·18-s + 0.148·19-s − 1.34·20-s − 0.218·21-s + 0.516·22-s + 0.605·23-s − 0.101·24-s + 1.38·25-s − 1.65·26-s + 0.192·27-s − 0.329·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.901749086\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.901749086\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 383 | \( 1 + T \) |
good | 2 | \( 1 - 1.93T + 2T^{2} \) |
| 5 | \( 1 + 3.45T + 5T^{2} \) |
| 11 | \( 1 - 1.25T + 11T^{2} \) |
| 13 | \( 1 + 4.36T + 13T^{2} \) |
| 17 | \( 1 - 0.181T + 17T^{2} \) |
| 19 | \( 1 - 0.648T + 19T^{2} \) |
| 23 | \( 1 - 2.90T + 23T^{2} \) |
| 29 | \( 1 + 1.43T + 29T^{2} \) |
| 31 | \( 1 - 9.47T + 31T^{2} \) |
| 37 | \( 1 - 2.00T + 37T^{2} \) |
| 41 | \( 1 + 8.98T + 41T^{2} \) |
| 43 | \( 1 + 8.24T + 43T^{2} \) |
| 47 | \( 1 - 10.2T + 47T^{2} \) |
| 53 | \( 1 - 3.61T + 53T^{2} \) |
| 59 | \( 1 - 5.98T + 59T^{2} \) |
| 61 | \( 1 - 13.2T + 61T^{2} \) |
| 67 | \( 1 - 4.90T + 67T^{2} \) |
| 71 | \( 1 - 0.491T + 71T^{2} \) |
| 73 | \( 1 - 13.1T + 73T^{2} \) |
| 79 | \( 1 + 14.7T + 79T^{2} \) |
| 83 | \( 1 - 5.31T + 83T^{2} \) |
| 89 | \( 1 - 16.2T + 89T^{2} \) |
| 97 | \( 1 + 0.676T + 97T^{2} \) |
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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
−7.73653697467250453324602384781, −6.86042410323570452666426329870, −6.71225366682488945817188388525, −5.43422977364508620966563060800, −4.83515921668083204270357233905, −4.18676232580560436137788076403, −3.63389564606244944700169588462, −3.01912251767057446644531776616, −2.30125984469083712091044220947, −0.63695715617002291793213612606,
0.63695715617002291793213612606, 2.30125984469083712091044220947, 3.01912251767057446644531776616, 3.63389564606244944700169588462, 4.18676232580560436137788076403, 4.83515921668083204270357233905, 5.43422977364508620966563060800, 6.71225366682488945817188388525, 6.86042410323570452666426329870, 7.73653697467250453324602384781