L(s) = 1 | − 2.41·2-s + 3-s + 3.82·4-s − 2·5-s − 2.41·6-s − 4.41·8-s + 9-s + 4.82·10-s + 11-s + 3.82·12-s − 0.828·13-s − 2·15-s + 2.99·16-s − 4.41·17-s − 2.41·18-s + 7.24·19-s − 7.65·20-s − 2.41·22-s − 7·23-s − 4.41·24-s − 25-s + 1.99·26-s + 27-s + 3.24·29-s + 4.82·30-s + 5.65·31-s + 1.58·32-s + ⋯ |
L(s) = 1 | − 1.70·2-s + 0.577·3-s + 1.91·4-s − 0.894·5-s − 0.985·6-s − 1.56·8-s + 0.333·9-s + 1.52·10-s + 0.301·11-s + 1.10·12-s − 0.229·13-s − 0.516·15-s + 0.749·16-s − 1.07·17-s − 0.569·18-s + 1.66·19-s − 1.71·20-s − 0.514·22-s − 1.45·23-s − 0.901·24-s − 0.200·25-s + 0.392·26-s + 0.192·27-s + 0.602·29-s + 0.881·30-s + 1.01·31-s + 0.280·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 - T \) |
good | 2 | \( 1 + 2.41T + 2T^{2} \) |
| 5 | \( 1 + 2T + 5T^{2} \) |
| 13 | \( 1 + 0.828T + 13T^{2} \) |
| 17 | \( 1 + 4.41T + 17T^{2} \) |
| 19 | \( 1 - 7.24T + 19T^{2} \) |
| 23 | \( 1 + 7T + 23T^{2} \) |
| 29 | \( 1 - 3.24T + 29T^{2} \) |
| 31 | \( 1 - 5.65T + 31T^{2} \) |
| 37 | \( 1 + 9.48T + 37T^{2} \) |
| 41 | \( 1 + 1.17T + 41T^{2} \) |
| 43 | \( 1 + 2.75T + 43T^{2} \) |
| 47 | \( 1 - 9.82T + 47T^{2} \) |
| 53 | \( 1 + 7.17T + 53T^{2} \) |
| 59 | \( 1 + 8.65T + 59T^{2} \) |
| 61 | \( 1 + 4T + 61T^{2} \) |
| 67 | \( 1 + 3.17T + 67T^{2} \) |
| 71 | \( 1 + 4.17T + 71T^{2} \) |
| 73 | \( 1 - 0.343T + 73T^{2} \) |
| 79 | \( 1 - 13.3T + 79T^{2} \) |
| 83 | \( 1 + 2.82T + 83T^{2} \) |
| 89 | \( 1 + 14.1T + 89T^{2} \) |
| 97 | \( 1 + 11.4T + 97T^{2} \) |
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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
−8.925244273161343507784713384625, −8.274631162935441020542379527799, −7.66628361659559764926271557384, −7.08900222400342542256647763104, −6.18925706956845655307866109056, −4.69304038187189757170150256612, −3.63038134202443877586541385678, −2.55269324795494843239122022897, −1.41942960474794095224470410930, 0,
1.41942960474794095224470410930, 2.55269324795494843239122022897, 3.63038134202443877586541385678, 4.69304038187189757170150256612, 6.18925706956845655307866109056, 7.08900222400342542256647763104, 7.66628361659559764926271557384, 8.274631162935441020542379527799, 8.925244273161343507784713384625