L(s) = 1 | + 1.64·2-s + 1.49·3-s + 0.706·4-s − 5-s + 2.45·6-s − 7-s − 2.12·8-s − 0.768·9-s − 1.64·10-s + 3.95·11-s + 1.05·12-s − 0.828·13-s − 1.64·14-s − 1.49·15-s − 4.91·16-s + 3.79·17-s − 1.26·18-s − 5.95·19-s − 0.706·20-s − 1.49·21-s + 6.50·22-s + 7.48·23-s − 3.17·24-s + 25-s − 1.36·26-s − 5.62·27-s − 0.706·28-s + ⋯ |
L(s) = 1 | + 1.16·2-s + 0.862·3-s + 0.353·4-s − 0.447·5-s + 1.00·6-s − 0.377·7-s − 0.752·8-s − 0.256·9-s − 0.520·10-s + 1.19·11-s + 0.304·12-s − 0.229·13-s − 0.439·14-s − 0.385·15-s − 1.22·16-s + 0.919·17-s − 0.297·18-s − 1.36·19-s − 0.157·20-s − 0.325·21-s + 1.38·22-s + 1.56·23-s − 0.648·24-s + 0.200·25-s − 0.267·26-s − 1.08·27-s − 0.133·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 229 | \( 1 + T \) |
good | 2 | \( 1 - 1.64T + 2T^{2} \) |
| 3 | \( 1 - 1.49T + 3T^{2} \) |
| 11 | \( 1 - 3.95T + 11T^{2} \) |
| 13 | \( 1 + 0.828T + 13T^{2} \) |
| 17 | \( 1 - 3.79T + 17T^{2} \) |
| 19 | \( 1 + 5.95T + 19T^{2} \) |
| 23 | \( 1 - 7.48T + 23T^{2} \) |
| 29 | \( 1 - 5.22T + 29T^{2} \) |
| 31 | \( 1 + 7.81T + 31T^{2} \) |
| 37 | \( 1 + 3.01T + 37T^{2} \) |
| 41 | \( 1 + 3.04T + 41T^{2} \) |
| 43 | \( 1 - 0.859T + 43T^{2} \) |
| 47 | \( 1 + 7.61T + 47T^{2} \) |
| 53 | \( 1 + 8.86T + 53T^{2} \) |
| 59 | \( 1 + 7.24T + 59T^{2} \) |
| 61 | \( 1 - 0.609T + 61T^{2} \) |
| 67 | \( 1 - 8.30T + 67T^{2} \) |
| 71 | \( 1 + 9.37T + 71T^{2} \) |
| 73 | \( 1 + 8.32T + 73T^{2} \) |
| 79 | \( 1 + 6.58T + 79T^{2} \) |
| 83 | \( 1 - 1.31T + 83T^{2} \) |
| 89 | \( 1 + 13.1T + 89T^{2} \) |
| 97 | \( 1 + 9.66T + 97T^{2} \) |
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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
−7.34332071382846164502293685527, −6.67188629861208802991450495277, −6.08276443543212944462341221781, −5.23599782868432867788245778493, −4.52228090411216060133428315185, −3.78962182761153867352515195474, −3.27165276238629520799451413437, −2.72308629011295269920487136516, −1.54353004334030019675952529494, 0,
1.54353004334030019675952529494, 2.72308629011295269920487136516, 3.27165276238629520799451413437, 3.78962182761153867352515195474, 4.52228090411216060133428315185, 5.23599782868432867788245778493, 6.08276443543212944462341221781, 6.67188629861208802991450495277, 7.34332071382846164502293685527