L(s) = 1 | + 1.26·2-s − 3.11·3-s − 0.404·4-s − 4.34·5-s − 3.93·6-s + 1.70·7-s − 3.03·8-s + 6.69·9-s − 5.48·10-s + 2.71·11-s + 1.25·12-s + 5.09·13-s + 2.15·14-s + 13.5·15-s − 3.02·16-s − 2.83·17-s + 8.45·18-s − 6.52·19-s + 1.75·20-s − 5.30·21-s + 3.43·22-s + 5.21·23-s + 9.45·24-s + 13.8·25-s + 6.44·26-s − 11.4·27-s − 0.688·28-s + ⋯ |
L(s) = 1 | + 0.893·2-s − 1.79·3-s − 0.202·4-s − 1.94·5-s − 1.60·6-s + 0.643·7-s − 1.07·8-s + 2.23·9-s − 1.73·10-s + 0.819·11-s + 0.363·12-s + 1.41·13-s + 0.574·14-s + 3.48·15-s − 0.756·16-s − 0.686·17-s + 1.99·18-s − 1.49·19-s + 0.392·20-s − 1.15·21-s + 0.732·22-s + 1.08·23-s + 1.93·24-s + 2.76·25-s + 1.26·26-s − 2.21·27-s − 0.130·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 503 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 503 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7215731187\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7215731187\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 503 | \( 1 - T \) |
good | 2 | \( 1 - 1.26T + 2T^{2} \) |
| 3 | \( 1 + 3.11T + 3T^{2} \) |
| 5 | \( 1 + 4.34T + 5T^{2} \) |
| 7 | \( 1 - 1.70T + 7T^{2} \) |
| 11 | \( 1 - 2.71T + 11T^{2} \) |
| 13 | \( 1 - 5.09T + 13T^{2} \) |
| 17 | \( 1 + 2.83T + 17T^{2} \) |
| 19 | \( 1 + 6.52T + 19T^{2} \) |
| 23 | \( 1 - 5.21T + 23T^{2} \) |
| 29 | \( 1 + 2.54T + 29T^{2} \) |
| 31 | \( 1 - 4.54T + 31T^{2} \) |
| 37 | \( 1 + 1.18T + 37T^{2} \) |
| 41 | \( 1 - 7.99T + 41T^{2} \) |
| 43 | \( 1 - 3.86T + 43T^{2} \) |
| 47 | \( 1 + 2.23T + 47T^{2} \) |
| 53 | \( 1 + 3.66T + 53T^{2} \) |
| 59 | \( 1 + 4.73T + 59T^{2} \) |
| 61 | \( 1 - 14.8T + 61T^{2} \) |
| 67 | \( 1 - 8.49T + 67T^{2} \) |
| 71 | \( 1 - 1.02T + 71T^{2} \) |
| 73 | \( 1 - 9.72T + 73T^{2} \) |
| 79 | \( 1 - 1.31T + 79T^{2} \) |
| 83 | \( 1 - 6.92T + 83T^{2} \) |
| 89 | \( 1 + 8.39T + 89T^{2} \) |
| 97 | \( 1 - 5.39T + 97T^{2} \) |
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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
−11.31814169308892186537328867918, −10.76442900198117901505444301907, −8.986114744219924518856519414767, −8.152965804686454571952182701255, −6.82205594585256713142842714470, −6.23922249903565109194727495751, −4.97764885847312103880356030271, −4.31834457130233026455774077564, −3.75933349499857200731719149276, −0.75205671956703505828484342844,
0.75205671956703505828484342844, 3.75933349499857200731719149276, 4.31834457130233026455774077564, 4.97764885847312103880356030271, 6.23922249903565109194727495751, 6.82205594585256713142842714470, 8.152965804686454571952182701255, 8.986114744219924518856519414767, 10.76442900198117901505444301907, 11.31814169308892186537328867918