L(s) = 1 | − 2.20·2-s − 1.74·3-s + 2.88·4-s + 1.63·5-s + 3.86·6-s + 4.01·7-s − 1.95·8-s + 0.0577·9-s − 3.60·10-s − 3.13·11-s − 5.04·12-s − 2.02·13-s − 8.86·14-s − 2.85·15-s − 1.45·16-s − 1.32·17-s − 0.127·18-s − 19-s + 4.70·20-s − 7.01·21-s + 6.92·22-s − 0.160·23-s + 3.41·24-s − 2.33·25-s + 4.47·26-s + 5.14·27-s + 11.5·28-s + ⋯ |
L(s) = 1 | − 1.56·2-s − 1.00·3-s + 1.44·4-s + 0.729·5-s + 1.57·6-s + 1.51·7-s − 0.690·8-s + 0.0192·9-s − 1.14·10-s − 0.944·11-s − 1.45·12-s − 0.561·13-s − 2.36·14-s − 0.736·15-s − 0.363·16-s − 0.321·17-s − 0.0300·18-s − 0.229·19-s + 1.05·20-s − 1.53·21-s + 1.47·22-s − 0.0334·23-s + 0.696·24-s − 0.467·25-s + 0.877·26-s + 0.990·27-s + 2.18·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6023 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 19 | \( 1 + T \) |
| 317 | \( 1 + T \) |
good | 2 | \( 1 + 2.20T + 2T^{2} \) |
| 3 | \( 1 + 1.74T + 3T^{2} \) |
| 5 | \( 1 - 1.63T + 5T^{2} \) |
| 7 | \( 1 - 4.01T + 7T^{2} \) |
| 11 | \( 1 + 3.13T + 11T^{2} \) |
| 13 | \( 1 + 2.02T + 13T^{2} \) |
| 17 | \( 1 + 1.32T + 17T^{2} \) |
| 23 | \( 1 + 0.160T + 23T^{2} \) |
| 29 | \( 1 + 3.17T + 29T^{2} \) |
| 31 | \( 1 - 2.56T + 31T^{2} \) |
| 37 | \( 1 - 2.99T + 37T^{2} \) |
| 41 | \( 1 - 9.36T + 41T^{2} \) |
| 43 | \( 1 + 3.77T + 43T^{2} \) |
| 47 | \( 1 + 2.24T + 47T^{2} \) |
| 53 | \( 1 - 8.94T + 53T^{2} \) |
| 59 | \( 1 - 9.63T + 59T^{2} \) |
| 61 | \( 1 + 11.0T + 61T^{2} \) |
| 67 | \( 1 - 0.661T + 67T^{2} \) |
| 71 | \( 1 + 8.70T + 71T^{2} \) |
| 73 | \( 1 - 0.404T + 73T^{2} \) |
| 79 | \( 1 + 13.1T + 79T^{2} \) |
| 83 | \( 1 - 4.71T + 83T^{2} \) |
| 89 | \( 1 + 5.22T + 89T^{2} \) |
| 97 | \( 1 - 3.17T + 97T^{2} \) |
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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
−7.71091462807231224187706304314, −7.37502448117231065755098615216, −6.34032158439508634755844841631, −5.69554343187900198395155135587, −5.02188588041992515741231115140, −4.38803224479635204087213031543, −2.59192222763023530274698689467, −2.00995573594091975354289566072, −1.07467665092555401357473050463, 0,
1.07467665092555401357473050463, 2.00995573594091975354289566072, 2.59192222763023530274698689467, 4.38803224479635204087213031543, 5.02188588041992515741231115140, 5.69554343187900198395155135587, 6.34032158439508634755844841631, 7.37502448117231065755098615216, 7.71091462807231224187706304314