L(s) = 1 | + 1.72·2-s + 0.981·4-s + 3.51·5-s − 1.75·8-s + 6.06·10-s + 6.09·11-s + 1.12·13-s − 4.99·16-s + 1.20·17-s + 2.20·19-s + 3.44·20-s + 10.5·22-s + 1.27·23-s + 7.33·25-s + 1.93·26-s − 6.20·29-s + 0.188·31-s − 5.11·32-s + 2.07·34-s + 3.57·37-s + 3.80·38-s − 6.17·40-s − 3.36·41-s + 3.80·43-s + 5.97·44-s + 2.19·46-s − 5.72·47-s + ⋯ |
L(s) = 1 | + 1.22·2-s + 0.490·4-s + 1.57·5-s − 0.621·8-s + 1.91·10-s + 1.83·11-s + 0.310·13-s − 1.24·16-s + 0.292·17-s + 0.505·19-s + 0.770·20-s + 2.24·22-s + 0.265·23-s + 1.46·25-s + 0.379·26-s − 1.15·29-s + 0.0338·31-s − 0.904·32-s + 0.356·34-s + 0.588·37-s + 0.617·38-s − 0.976·40-s − 0.525·41-s + 0.580·43-s + 0.900·44-s + 0.324·46-s − 0.834·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.210751868\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.210751868\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 - 1.72T + 2T^{2} \) |
| 5 | \( 1 - 3.51T + 5T^{2} \) |
| 11 | \( 1 - 6.09T + 11T^{2} \) |
| 13 | \( 1 - 1.12T + 13T^{2} \) |
| 17 | \( 1 - 1.20T + 17T^{2} \) |
| 19 | \( 1 - 2.20T + 19T^{2} \) |
| 23 | \( 1 - 1.27T + 23T^{2} \) |
| 29 | \( 1 + 6.20T + 29T^{2} \) |
| 31 | \( 1 - 0.188T + 31T^{2} \) |
| 37 | \( 1 - 3.57T + 37T^{2} \) |
| 41 | \( 1 + 3.36T + 41T^{2} \) |
| 43 | \( 1 - 3.80T + 43T^{2} \) |
| 47 | \( 1 + 5.72T + 47T^{2} \) |
| 53 | \( 1 - 8.33T + 53T^{2} \) |
| 59 | \( 1 + 11.2T + 59T^{2} \) |
| 61 | \( 1 + 12.0T + 61T^{2} \) |
| 67 | \( 1 + 7.91T + 67T^{2} \) |
| 71 | \( 1 - 12.2T + 71T^{2} \) |
| 73 | \( 1 - 5.31T + 73T^{2} \) |
| 79 | \( 1 - 9.21T + 79T^{2} \) |
| 83 | \( 1 + 1.24T + 83T^{2} \) |
| 89 | \( 1 - 5.54T + 89T^{2} \) |
| 97 | \( 1 - 16.4T + 97T^{2} \) |
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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
−8.736786735756355049858218415849, −7.45001184212808657546227433384, −6.44587813446745896523289378537, −6.21166068580606825451172130021, −5.48664912514771909926033961854, −4.77189446343632461377530162393, −3.86596669449949432842186909153, −3.18788645833983328454135988117, −2.10350034172108527560169417578, −1.22802909591996836046630465227,
1.22802909591996836046630465227, 2.10350034172108527560169417578, 3.18788645833983328454135988117, 3.86596669449949432842186909153, 4.77189446343632461377530162393, 5.48664912514771909926033961854, 6.21166068580606825451172130021, 6.44587813446745896523289378537, 7.45001184212808657546227433384, 8.736786735756355049858218415849