| L(s) = 1 | + (−1 + i)2-s + i·3-s − 2i·4-s + (−2 + 2i)5-s + (−1 − i)6-s − 2i·7-s + (2 + 2i)8-s − 9-s − 4i·10-s + 4i·11-s + 2·12-s + (−1 + i)13-s + (2 + 2i)14-s + (−2 − 2i)15-s − 4·16-s + (−2 − 2i)17-s + ⋯ |
| L(s) = 1 | + (−0.707 + 0.707i)2-s + 0.577i·3-s − i·4-s + (−0.894 + 0.894i)5-s + (−0.408 − 0.408i)6-s − 0.755i·7-s + (0.707 + 0.707i)8-s − 0.333·9-s − 1.26i·10-s + 1.20i·11-s + 0.577·12-s + (−0.277 + 0.277i)13-s + (0.534 + 0.534i)14-s + (−0.516 − 0.516i)15-s − 16-s + (−0.485 − 0.485i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0824 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0824 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
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 | \( 1 + (1 - i)T \) |
| 3 | \( 1 - iT \) |
| 37 | \( 1 + (1 + 6i)T \) |
| good | 5 | \( 1 + (2 - 2i)T - 5iT^{2} \) |
| 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 - 4iT - 11T^{2} \) |
| 13 | \( 1 + (1 - i)T - 13iT^{2} \) |
| 17 | \( 1 + (2 + 2i)T + 17iT^{2} \) |
| 19 | \( 1 + (-1 - i)T + 19iT^{2} \) |
| 23 | \( 1 + (2 - 2i)T - 23iT^{2} \) |
| 29 | \( 1 + (2 + 2i)T + 29iT^{2} \) |
| 31 | \( 1 + (7 + 7i)T + 31iT^{2} \) |
| 41 | \( 1 + 2iT - 41T^{2} \) |
| 43 | \( 1 + (-3 - 3i)T + 43iT^{2} \) |
| 47 | \( 1 + 4iT - 47T^{2} \) |
| 53 | \( 1 + 2iT - 53T^{2} \) |
| 59 | \( 1 + (8 + 8i)T + 59iT^{2} \) |
| 61 | \( 1 + (1 + i)T + 61iT^{2} \) |
| 67 | \( 1 + 2iT - 67T^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 - 6iT - 73T^{2} \) |
| 79 | \( 1 + (3 - 3i)T - 79iT^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 - 89iT^{2} \) |
| 97 | \( 1 + (-3 - 3i)T + 97iT^{2} \) |
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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
−9.738633219224020670869408153345, −9.321924135994664025758133813425, −7.981126261820416233182584945004, −7.34847979860524319173576549594, −6.91534628999212558492856793582, −5.66648279770162576168985679613, −4.50867558539335250968776920496, −3.78090169944807331430310030282, −2.15333976409699628574921211347, 0,
1.33002203539019968278374339651, 2.72766549364153072777069575179, 3.73035769981333388573314648440, 4.91607225912987517143439621329, 6.06767403421371306206175065841, 7.27314193888750601323625848807, 8.099025046418269072891519114432, 8.720641910488239256617846959335, 9.108133118196141956323796647472
