L(s) = 1 | − 3-s − 2·4-s − 1.58·5-s − 3.08·7-s + 9-s + 6.03·11-s + 2·12-s − 2.36·13-s + 1.58·15-s + 4·16-s + 6.85·17-s − 5.13·19-s + 3.16·20-s + 3.08·21-s + 23-s − 2.50·25-s − 27-s + 6.16·28-s + 29-s + 0.0784·31-s − 6.03·33-s + 4.87·35-s − 2·36-s + 5.42·37-s + 2.36·39-s + 7.95·41-s − 2.05·43-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 4-s − 0.706·5-s − 1.16·7-s + 0.333·9-s + 1.81·11-s + 0.577·12-s − 0.657·13-s + 0.408·15-s + 16-s + 1.66·17-s − 1.17·19-s + 0.706·20-s + 0.672·21-s + 0.208·23-s − 0.500·25-s − 0.192·27-s + 1.16·28-s + 0.185·29-s + 0.0140·31-s − 1.05·33-s + 0.823·35-s − 0.333·36-s + 0.892·37-s + 0.379·39-s + 1.24·41-s − 0.313·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
good | 2 | \( 1 + 2T^{2} \) |
| 5 | \( 1 + 1.58T + 5T^{2} \) |
| 7 | \( 1 + 3.08T + 7T^{2} \) |
| 11 | \( 1 - 6.03T + 11T^{2} \) |
| 13 | \( 1 + 2.36T + 13T^{2} \) |
| 17 | \( 1 - 6.85T + 17T^{2} \) |
| 19 | \( 1 + 5.13T + 19T^{2} \) |
| 31 | \( 1 - 0.0784T + 31T^{2} \) |
| 37 | \( 1 - 5.42T + 37T^{2} \) |
| 41 | \( 1 - 7.95T + 41T^{2} \) |
| 43 | \( 1 + 2.05T + 43T^{2} \) |
| 47 | \( 1 - 1.58T + 47T^{2} \) |
| 53 | \( 1 + 5.00T + 53T^{2} \) |
| 59 | \( 1 + 0.756T + 59T^{2} \) |
| 61 | \( 1 + 3.79T + 61T^{2} \) |
| 67 | \( 1 - 6.16T + 67T^{2} \) |
| 71 | \( 1 + 14.1T + 71T^{2} \) |
| 73 | \( 1 + 2.91T + 73T^{2} \) |
| 79 | \( 1 + 14.1T + 79T^{2} \) |
| 83 | \( 1 - 5.48T + 83T^{2} \) |
| 89 | \( 1 + 13.5T + 89T^{2} \) |
| 97 | \( 1 - 2.53T + 97T^{2} \) |
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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
−8.918325944322273856967751359152, −7.992981097090725045461239846005, −7.17001901798670917947504647815, −6.28611956699328826844239477422, −5.68500440782437704851733296548, −4.43296074746332024618418378238, −3.98218699978229777054595998930, −3.12061721193960262082869744068, −1.18040099070440492574044301542, 0,
1.18040099070440492574044301542, 3.12061721193960262082869744068, 3.98218699978229777054595998930, 4.43296074746332024618418378238, 5.68500440782437704851733296548, 6.28611956699328826844239477422, 7.17001901798670917947504647815, 7.992981097090725045461239846005, 8.918325944322273856967751359152