| L(s) = 1 | − 1.81i·3-s − 2.70i·5-s + 2.58·7-s − 0.298·9-s + 0.772i·11-s − i·13-s − 4.90·15-s − 0.701·17-s − 4.40i·19-s − 4.70i·21-s + 3.63·23-s − 2.29·25-s − 4.90i·27-s − 2i·29-s + 5.95·31-s + ⋯ |
| L(s) = 1 | − 1.04i·3-s − 1.20i·5-s + 0.978·7-s − 0.0994·9-s + 0.232i·11-s − 0.277i·13-s − 1.26·15-s − 0.170·17-s − 1.01i·19-s − 1.02i·21-s + 0.757·23-s − 0.459·25-s − 0.944i·27-s − 0.371i·29-s + 1.06·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1664 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.932298912\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.932298912\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 + iT \) |
| good | 3 | \( 1 + 1.81iT - 3T^{2} \) |
| 5 | \( 1 + 2.70iT - 5T^{2} \) |
| 7 | \( 1 - 2.58T + 7T^{2} \) |
| 11 | \( 1 - 0.772iT - 11T^{2} \) |
| 17 | \( 1 + 0.701T + 17T^{2} \) |
| 19 | \( 1 + 4.40iT - 19T^{2} \) |
| 23 | \( 1 - 3.63T + 23T^{2} \) |
| 29 | \( 1 + 2iT - 29T^{2} \) |
| 31 | \( 1 - 5.95T + 31T^{2} \) |
| 37 | \( 1 + 6.70iT - 37T^{2} \) |
| 41 | \( 1 + 11.4T + 41T^{2} \) |
| 43 | \( 1 - 10.0iT - 43T^{2} \) |
| 47 | \( 1 - 9.31T + 47T^{2} \) |
| 53 | \( 1 - 1.40iT - 53T^{2} \) |
| 59 | \( 1 - 2.85iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 12.6iT - 67T^{2} \) |
| 71 | \( 1 + 1.04T + 71T^{2} \) |
| 73 | \( 1 + 7.40T + 73T^{2} \) |
| 79 | \( 1 + 2.08T + 79T^{2} \) |
| 83 | \( 1 + 9.58iT - 83T^{2} \) |
| 89 | \( 1 + 11.4T + 89T^{2} \) |
| 97 | \( 1 - 10T + 97T^{2} \) |
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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
−8.779034345790656712794182224398, −8.308534798931805449986328520568, −7.50283223636639314404368651419, −6.85028579680785742707735090957, −5.77762343213665783379286193078, −4.85730947131298683905851441046, −4.35293151883386116980407098943, −2.67461910804919592598088679456, −1.57850004246862123975302697670, −0.802001423916207920060838061641,
1.65468930621483847090460952823, 2.96020137619498400375249156727, 3.75584890588958472980431483308, 4.66479884736988741823673066360, 5.40470211050028906063053057866, 6.53516742399342908874998953220, 7.20037112143827759330198672870, 8.182716864639131448977987811402, 8.884472652395008819484926622079, 9.930239068120496568384984455077
