L(s) = 1 | − 2.41·2-s − 3-s + 3.82·4-s − 5-s + 2.41·6-s − 1.41·7-s − 4.41·8-s + 9-s + 2.41·10-s − 2.24·11-s − 3.82·12-s + 3.41·13-s + 3.41·14-s + 15-s + 2.99·16-s + 1.17·17-s − 2.41·18-s − 3.82·20-s + 1.41·21-s + 5.41·22-s + 7.65·23-s + 4.41·24-s + 25-s − 8.24·26-s − 27-s − 5.41·28-s − 1.41·29-s + ⋯ |
L(s) = 1 | − 1.70·2-s − 0.577·3-s + 1.91·4-s − 0.447·5-s + 0.985·6-s − 0.534·7-s − 1.56·8-s + 0.333·9-s + 0.763·10-s − 0.676·11-s − 1.10·12-s + 0.946·13-s + 0.912·14-s + 0.258·15-s + 0.749·16-s + 0.284·17-s − 0.569·18-s − 0.856·20-s + 0.308·21-s + 1.15·22-s + 1.59·23-s + 0.901·24-s + 0.200·25-s − 1.61·26-s − 0.192·27-s − 1.02·28-s − 0.262·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5415 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5415 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4988150845\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4988150845\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + 2.41T + 2T^{2} \) |
| 7 | \( 1 + 1.41T + 7T^{2} \) |
| 11 | \( 1 + 2.24T + 11T^{2} \) |
| 13 | \( 1 - 3.41T + 13T^{2} \) |
| 17 | \( 1 - 1.17T + 17T^{2} \) |
| 23 | \( 1 - 7.65T + 23T^{2} \) |
| 29 | \( 1 + 1.41T + 29T^{2} \) |
| 31 | \( 1 - 3.17T + 31T^{2} \) |
| 37 | \( 1 - 3.41T + 37T^{2} \) |
| 41 | \( 1 - 0.242T + 41T^{2} \) |
| 43 | \( 1 - 12.2T + 43T^{2} \) |
| 47 | \( 1 + 7.65T + 47T^{2} \) |
| 53 | \( 1 + 8T + 53T^{2} \) |
| 59 | \( 1 + 12.4T + 59T^{2} \) |
| 61 | \( 1 - 7.31T + 61T^{2} \) |
| 67 | \( 1 - 9.65T + 67T^{2} \) |
| 71 | \( 1 - 10.8T + 71T^{2} \) |
| 73 | \( 1 + 7.65T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 12.8T + 83T^{2} \) |
| 89 | \( 1 + 5.89T + 89T^{2} \) |
| 97 | \( 1 - 9.75T + 97T^{2} \) |
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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
−7.961297822617072408070681644727, −7.84721208835245750560075965943, −6.79235367677849492814735616609, −6.44410648581459075230236521785, −5.50432090076571703867907081640, −4.57090435077098425679869837570, −3.41621185245044703199838105347, −2.61154525792176734515021869087, −1.35762270765441350899846327149, −0.55654924884430229056701803874,
0.55654924884430229056701803874, 1.35762270765441350899846327149, 2.61154525792176734515021869087, 3.41621185245044703199838105347, 4.57090435077098425679869837570, 5.50432090076571703867907081640, 6.44410648581459075230236521785, 6.79235367677849492814735616609, 7.84721208835245750560075965943, 7.961297822617072408070681644727