L(s) = 1 | + 1.49·2-s − 1.13·3-s + 0.240·4-s + 3.48·5-s − 1.69·6-s − 1.05·7-s − 2.63·8-s − 1.71·9-s + 5.21·10-s + 5.79·11-s − 0.272·12-s + 2.73·13-s − 1.58·14-s − 3.95·15-s − 4.42·16-s − 17-s − 2.55·18-s + 4.97·19-s + 0.836·20-s + 1.19·21-s + 8.67·22-s + 3.90·23-s + 2.99·24-s + 7.11·25-s + 4.09·26-s + 5.34·27-s − 0.253·28-s + ⋯ |
L(s) = 1 | + 1.05·2-s − 0.655·3-s + 0.120·4-s + 1.55·5-s − 0.693·6-s − 0.399·7-s − 0.931·8-s − 0.570·9-s + 1.64·10-s + 1.74·11-s − 0.0787·12-s + 0.759·13-s − 0.422·14-s − 1.02·15-s − 1.10·16-s − 0.242·17-s − 0.603·18-s + 1.14·19-s + 0.187·20-s + 0.261·21-s + 1.84·22-s + 0.814·23-s + 0.610·24-s + 1.42·25-s + 0.804·26-s + 1.02·27-s − 0.0479·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.604049770\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.604049770\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 17 | \( 1 + T \) |
| 59 | \( 1 - T \) |
good | 2 | \( 1 - 1.49T + 2T^{2} \) |
| 3 | \( 1 + 1.13T + 3T^{2} \) |
| 5 | \( 1 - 3.48T + 5T^{2} \) |
| 7 | \( 1 + 1.05T + 7T^{2} \) |
| 11 | \( 1 - 5.79T + 11T^{2} \) |
| 13 | \( 1 - 2.73T + 13T^{2} \) |
| 19 | \( 1 - 4.97T + 19T^{2} \) |
| 23 | \( 1 - 3.90T + 23T^{2} \) |
| 29 | \( 1 - 0.0187T + 29T^{2} \) |
| 31 | \( 1 + 5.26T + 31T^{2} \) |
| 37 | \( 1 - 10.5T + 37T^{2} \) |
| 41 | \( 1 - 11.1T + 41T^{2} \) |
| 43 | \( 1 + 12.9T + 43T^{2} \) |
| 47 | \( 1 + 6.43T + 47T^{2} \) |
| 53 | \( 1 - 3.67T + 53T^{2} \) |
| 61 | \( 1 + 4.10T + 61T^{2} \) |
| 67 | \( 1 - 12.3T + 67T^{2} \) |
| 71 | \( 1 - 0.517T + 71T^{2} \) |
| 73 | \( 1 - 10.8T + 73T^{2} \) |
| 79 | \( 1 + 12.1T + 79T^{2} \) |
| 83 | \( 1 + 13.5T + 83T^{2} \) |
| 89 | \( 1 + 6.25T + 89T^{2} \) |
| 97 | \( 1 - 2.06T + 97T^{2} \) |
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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
−9.722026497237046914805258011207, −9.385704224698776738011074519957, −8.604088598153754256149716347661, −6.77404029427285267861193751652, −6.25565557799727912491276167634, −5.70356386017617261725151422820, −4.95218957437531948908504735048, −3.77383312388777461739416007636, −2.80334750711685515540162806082, −1.24279429723016255679406153848,
1.24279429723016255679406153848, 2.80334750711685515540162806082, 3.77383312388777461739416007636, 4.95218957437531948908504735048, 5.70356386017617261725151422820, 6.25565557799727912491276167634, 6.77404029427285267861193751652, 8.604088598153754256149716347661, 9.385704224698776738011074519957, 9.722026497237046914805258011207