L(s) = 1 | − 2-s + 2.23·3-s + 4-s + 4.33·5-s − 2.23·6-s + 3.84·7-s − 8-s + 1.99·9-s − 4.33·10-s + 0.985·11-s + 2.23·12-s − 5.10·13-s − 3.84·14-s + 9.68·15-s + 16-s + 3.80·17-s − 1.99·18-s − 2.32·19-s + 4.33·20-s + 8.58·21-s − 0.985·22-s + 6.44·23-s − 2.23·24-s + 13.7·25-s + 5.10·26-s − 2.23·27-s + 3.84·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.29·3-s + 0.5·4-s + 1.93·5-s − 0.912·6-s + 1.45·7-s − 0.353·8-s + 0.666·9-s − 1.36·10-s + 0.297·11-s + 0.645·12-s − 1.41·13-s − 1.02·14-s + 2.49·15-s + 0.250·16-s + 0.923·17-s − 0.471·18-s − 0.532·19-s + 0.968·20-s + 1.87·21-s − 0.210·22-s + 1.34·23-s − 0.456·24-s + 2.75·25-s + 1.00·26-s − 0.430·27-s + 0.725·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.154253396\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.154253396\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| 97 | \( 1 + T \) |
good | 3 | \( 1 - 2.23T + 3T^{2} \) |
| 5 | \( 1 - 4.33T + 5T^{2} \) |
| 7 | \( 1 - 3.84T + 7T^{2} \) |
| 11 | \( 1 - 0.985T + 11T^{2} \) |
| 13 | \( 1 + 5.10T + 13T^{2} \) |
| 17 | \( 1 - 3.80T + 17T^{2} \) |
| 19 | \( 1 + 2.32T + 19T^{2} \) |
| 23 | \( 1 - 6.44T + 23T^{2} \) |
| 29 | \( 1 + 2.94T + 29T^{2} \) |
| 37 | \( 1 - 2.66T + 37T^{2} \) |
| 41 | \( 1 + 7.12T + 41T^{2} \) |
| 43 | \( 1 + 8.08T + 43T^{2} \) |
| 47 | \( 1 - 12.2T + 47T^{2} \) |
| 53 | \( 1 + 7.50T + 53T^{2} \) |
| 59 | \( 1 - 1.69T + 59T^{2} \) |
| 61 | \( 1 - 6.35T + 61T^{2} \) |
| 67 | \( 1 + 12.8T + 67T^{2} \) |
| 71 | \( 1 - 12.7T + 71T^{2} \) |
| 73 | \( 1 + 10.9T + 73T^{2} \) |
| 79 | \( 1 + 9.16T + 79T^{2} \) |
| 83 | \( 1 + 6.75T + 83T^{2} \) |
| 89 | \( 1 + 1.26T + 89T^{2} \) |
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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
−8.361818809407391568246671366207, −7.42911078832195126395522073882, −7.01746159655940008529220213357, −5.92254338406956950897284899727, −5.24792958099871962131745724174, −4.60757772549431037857391423168, −3.17802720863603767470020263592, −2.47709449841744746010852108818, −1.89144847306440216684034895155, −1.26932283895849158306296474051,
1.26932283895849158306296474051, 1.89144847306440216684034895155, 2.47709449841744746010852108818, 3.17802720863603767470020263592, 4.60757772549431037857391423168, 5.24792958099871962131745724174, 5.92254338406956950897284899727, 7.01746159655940008529220213357, 7.42911078832195126395522073882, 8.361818809407391568246671366207