L(s) = 1 | + 2-s − 0.381·3-s + 4-s − 3·5-s − 0.381·6-s − 3.61·7-s + 8-s − 2.85·9-s − 3·10-s − 0.381·11-s − 0.381·12-s − 6.23·13-s − 3.61·14-s + 1.14·15-s + 16-s − 5·17-s − 2.85·18-s − 6.61·19-s − 3·20-s + 1.38·21-s − 0.381·22-s − 6.23·23-s − 0.381·24-s + 4·25-s − 6.23·26-s + 2.23·27-s − 3.61·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.220·3-s + 0.5·4-s − 1.34·5-s − 0.155·6-s − 1.36·7-s + 0.353·8-s − 0.951·9-s − 0.948·10-s − 0.115·11-s − 0.110·12-s − 1.72·13-s − 0.966·14-s + 0.295·15-s + 0.250·16-s − 1.21·17-s − 0.672·18-s − 1.51·19-s − 0.670·20-s + 0.301·21-s − 0.0814·22-s − 1.30·23-s − 0.0779·24-s + 0.800·25-s − 1.22·26-s + 0.430·27-s − 0.683·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 3023 | \( 1+O(T) \) |
good | 3 | \( 1 + 0.381T + 3T^{2} \) |
| 5 | \( 1 + 3T + 5T^{2} \) |
| 7 | \( 1 + 3.61T + 7T^{2} \) |
| 11 | \( 1 + 0.381T + 11T^{2} \) |
| 13 | \( 1 + 6.23T + 13T^{2} \) |
| 17 | \( 1 + 5T + 17T^{2} \) |
| 19 | \( 1 + 6.61T + 19T^{2} \) |
| 23 | \( 1 + 6.23T + 23T^{2} \) |
| 29 | \( 1 - 3.23T + 29T^{2} \) |
| 31 | \( 1 - 4.47T + 31T^{2} \) |
| 37 | \( 1 + 7.47T + 37T^{2} \) |
| 41 | \( 1 + 0.909T + 41T^{2} \) |
| 43 | \( 1 - 4.85T + 43T^{2} \) |
| 47 | \( 1 - 1.47T + 47T^{2} \) |
| 53 | \( 1 + 0.618T + 53T^{2} \) |
| 59 | \( 1 - 2.70T + 59T^{2} \) |
| 61 | \( 1 - 2.09T + 61T^{2} \) |
| 67 | \( 1 + 8T + 67T^{2} \) |
| 71 | \( 1 + 3.70T + 71T^{2} \) |
| 73 | \( 1 - 8.94T + 73T^{2} \) |
| 79 | \( 1 - 0.527T + 79T^{2} \) |
| 83 | \( 1 + 12.9T + 83T^{2} \) |
| 89 | \( 1 - 4.47T + 89T^{2} \) |
| 97 | \( 1 + 6.47T + 97T^{2} \) |
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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
−7.14986406050398670056444818843, −6.61815696621785750282684977632, −6.05030129321244386549756197301, −5.06646470461001858014284908912, −4.32846728345785024807493335882, −3.80269956502424158457407770484, −2.80207683302368755476818249028, −2.35127339045934623827273323973, 0, 0,
2.35127339045934623827273323973, 2.80207683302368755476818249028, 3.80269956502424158457407770484, 4.32846728345785024807493335882, 5.06646470461001858014284908912, 6.05030129321244386549756197301, 6.61815696621785750282684977632, 7.14986406050398670056444818843