L(s) = 1 | + 0.288·2-s − 3-s − 1.91·4-s + 5-s − 0.288·6-s − 0.447·7-s − 1.13·8-s + 9-s + 0.288·10-s − 3.45·11-s + 1.91·12-s + 5.09·13-s − 0.129·14-s − 15-s + 3.50·16-s − 3.00·17-s + 0.288·18-s − 6.49·19-s − 1.91·20-s + 0.447·21-s − 0.997·22-s − 0.708·23-s + 1.13·24-s + 25-s + 1.47·26-s − 27-s + 0.858·28-s + ⋯ |
L(s) = 1 | + 0.204·2-s − 0.577·3-s − 0.958·4-s + 0.447·5-s − 0.117·6-s − 0.169·7-s − 0.399·8-s + 0.333·9-s + 0.0913·10-s − 1.04·11-s + 0.553·12-s + 1.41·13-s − 0.0345·14-s − 0.258·15-s + 0.876·16-s − 0.727·17-s + 0.0680·18-s − 1.49·19-s − 0.428·20-s + 0.0977·21-s − 0.212·22-s − 0.147·23-s + 0.230·24-s + 0.200·25-s + 0.288·26-s − 0.192·27-s + 0.162·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9410509432\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9410509432\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 401 | \( 1 + T \) |
good | 2 | \( 1 - 0.288T + 2T^{2} \) |
| 7 | \( 1 + 0.447T + 7T^{2} \) |
| 11 | \( 1 + 3.45T + 11T^{2} \) |
| 13 | \( 1 - 5.09T + 13T^{2} \) |
| 17 | \( 1 + 3.00T + 17T^{2} \) |
| 19 | \( 1 + 6.49T + 19T^{2} \) |
| 23 | \( 1 + 0.708T + 23T^{2} \) |
| 29 | \( 1 + 6.50T + 29T^{2} \) |
| 31 | \( 1 + 0.921T + 31T^{2} \) |
| 37 | \( 1 - 3.37T + 37T^{2} \) |
| 41 | \( 1 - 4.39T + 41T^{2} \) |
| 43 | \( 1 - 2.87T + 43T^{2} \) |
| 47 | \( 1 - 11.0T + 47T^{2} \) |
| 53 | \( 1 - 5.23T + 53T^{2} \) |
| 59 | \( 1 - 1.40T + 59T^{2} \) |
| 61 | \( 1 + 9.14T + 61T^{2} \) |
| 67 | \( 1 + 10.5T + 67T^{2} \) |
| 71 | \( 1 - 1.07T + 71T^{2} \) |
| 73 | \( 1 - 3.14T + 73T^{2} \) |
| 79 | \( 1 + 5.49T + 79T^{2} \) |
| 83 | \( 1 - 3.57T + 83T^{2} \) |
| 89 | \( 1 - 0.624T + 89T^{2} \) |
| 97 | \( 1 - 0.895T + 97T^{2} \) |
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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
−8.178001558630848802502983878358, −7.38381850198596419680040392009, −6.32194190710019673037785551698, −5.91789322640259438919847260449, −5.31075212202259472466059862637, −4.36928371809585158612796127378, −3.95691513769609894443482247852, −2.83552621629943890314241823752, −1.77973099028443188417987758733, −0.50781199503161542161999514209,
0.50781199503161542161999514209, 1.77973099028443188417987758733, 2.83552621629943890314241823752, 3.95691513769609894443482247852, 4.36928371809585158612796127378, 5.31075212202259472466059862637, 5.91789322640259438919847260449, 6.32194190710019673037785551698, 7.38381850198596419680040392009, 8.178001558630848802502983878358