L(s) = 1 | + 2-s − 1.85·3-s + 4-s − 5-s − 1.85·6-s − 3.61·7-s + 8-s + 0.424·9-s − 10-s − 4.59·11-s − 1.85·12-s − 5.49·13-s − 3.61·14-s + 1.85·15-s + 16-s − 7.18·17-s + 0.424·18-s − 8.33·19-s − 20-s + 6.69·21-s − 4.59·22-s + 2.32·23-s − 1.85·24-s + 25-s − 5.49·26-s + 4.76·27-s − 3.61·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.06·3-s + 0.5·4-s − 0.447·5-s − 0.755·6-s − 1.36·7-s + 0.353·8-s + 0.141·9-s − 0.316·10-s − 1.38·11-s − 0.534·12-s − 1.52·13-s − 0.967·14-s + 0.477·15-s + 0.250·16-s − 1.74·17-s + 0.100·18-s − 1.91·19-s − 0.223·20-s + 1.46·21-s − 0.979·22-s + 0.484·23-s − 0.377·24-s + 0.200·25-s − 1.07·26-s + 0.917·27-s − 0.683·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1444612495\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1444612495\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 401 | \( 1 + T \) |
good | 3 | \( 1 + 1.85T + 3T^{2} \) |
| 7 | \( 1 + 3.61T + 7T^{2} \) |
| 11 | \( 1 + 4.59T + 11T^{2} \) |
| 13 | \( 1 + 5.49T + 13T^{2} \) |
| 17 | \( 1 + 7.18T + 17T^{2} \) |
| 19 | \( 1 + 8.33T + 19T^{2} \) |
| 23 | \( 1 - 2.32T + 23T^{2} \) |
| 29 | \( 1 + 0.581T + 29T^{2} \) |
| 31 | \( 1 - 2.15T + 31T^{2} \) |
| 37 | \( 1 - 0.265T + 37T^{2} \) |
| 41 | \( 1 + 1.58T + 41T^{2} \) |
| 43 | \( 1 - 9.22T + 43T^{2} \) |
| 47 | \( 1 - 6.13T + 47T^{2} \) |
| 53 | \( 1 - 8.25T + 53T^{2} \) |
| 59 | \( 1 + 0.454T + 59T^{2} \) |
| 61 | \( 1 - 1.76T + 61T^{2} \) |
| 67 | \( 1 + 5.09T + 67T^{2} \) |
| 71 | \( 1 + 12.0T + 71T^{2} \) |
| 73 | \( 1 + 6.51T + 73T^{2} \) |
| 79 | \( 1 + 15.9T + 79T^{2} \) |
| 83 | \( 1 + 0.411T + 83T^{2} \) |
| 89 | \( 1 + 3.77T + 89T^{2} \) |
| 97 | \( 1 - 11.9T + 97T^{2} \) |
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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
−8.418506976704505933555479516892, −7.28528126650085606859096000356, −6.87125438723522089473920894810, −6.14186760873614098649209175080, −5.51872575847951019118936397908, −4.62820927650849341063286105846, −4.19854751718338796115003153572, −2.74802496991185661200521086834, −2.47378860649158749375967152017, −0.18550417647918376145235661143,
0.18550417647918376145235661143, 2.47378860649158749375967152017, 2.74802496991185661200521086834, 4.19854751718338796115003153572, 4.62820927650849341063286105846, 5.51872575847951019118936397908, 6.14186760873614098649209175080, 6.87125438723522089473920894810, 7.28528126650085606859096000356, 8.418506976704505933555479516892