L(s) = 1 | + (−0.0980 + 0.995i)2-s + (−0.980 − 0.195i)4-s + (−0.956 + 0.290i)7-s + (0.290 − 0.956i)8-s + (−0.471 − 0.881i)9-s + (1.60 + 1.18i)11-s + (−0.195 − 0.980i)14-s + (0.923 + 0.382i)16-s + (0.923 − 0.382i)18-s + (−1.34 + 1.48i)22-s + (0.172 + 1.75i)23-s + (−0.773 + 0.634i)25-s + (0.995 − 0.0980i)28-s + (1.86 + 0.276i)29-s + (−0.471 + 0.881i)32-s + ⋯ |
L(s) = 1 | + (−0.0980 + 0.995i)2-s + (−0.980 − 0.195i)4-s + (−0.956 + 0.290i)7-s + (0.290 − 0.956i)8-s + (−0.471 − 0.881i)9-s + (1.60 + 1.18i)11-s + (−0.195 − 0.980i)14-s + (0.923 + 0.382i)16-s + (0.923 − 0.382i)18-s + (−1.34 + 1.48i)22-s + (0.172 + 1.75i)23-s + (−0.773 + 0.634i)25-s + (0.995 − 0.0980i)28-s + (1.86 + 0.276i)29-s + (−0.471 + 0.881i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.492 - 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.492 - 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8413682031\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8413682031\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.0980 - 0.995i)T \) |
| 7 | \( 1 + (0.956 - 0.290i)T \) |
good | 3 | \( 1 + (0.471 + 0.881i)T^{2} \) |
| 5 | \( 1 + (0.773 - 0.634i)T^{2} \) |
| 11 | \( 1 + (-1.60 - 1.18i)T + (0.290 + 0.956i)T^{2} \) |
| 13 | \( 1 + (0.634 - 0.773i)T^{2} \) |
| 17 | \( 1 + (-0.923 + 0.382i)T^{2} \) |
| 19 | \( 1 + (-0.995 + 0.0980i)T^{2} \) |
| 23 | \( 1 + (-0.172 - 1.75i)T + (-0.980 + 0.195i)T^{2} \) |
| 29 | \( 1 + (-1.86 - 0.276i)T + (0.956 + 0.290i)T^{2} \) |
| 31 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 37 | \( 1 + (1.19 - 1.07i)T + (0.0980 - 0.995i)T^{2} \) |
| 41 | \( 1 + (-0.195 - 0.980i)T^{2} \) |
| 43 | \( 1 + (-0.251 + 0.150i)T + (0.471 - 0.881i)T^{2} \) |
| 47 | \( 1 + (-0.382 - 0.923i)T^{2} \) |
| 53 | \( 1 + (1.91 - 0.284i)T + (0.956 - 0.290i)T^{2} \) |
| 59 | \( 1 + (0.634 + 0.773i)T^{2} \) |
| 61 | \( 1 + (0.881 - 0.471i)T^{2} \) |
| 67 | \( 1 + (-1.66 + 0.416i)T + (0.881 - 0.471i)T^{2} \) |
| 71 | \( 1 + (-0.979 - 0.523i)T + (0.555 + 0.831i)T^{2} \) |
| 73 | \( 1 + (-0.831 - 0.555i)T^{2} \) |
| 79 | \( 1 + (-0.704 - 1.05i)T + (-0.382 + 0.923i)T^{2} \) |
| 83 | \( 1 + (-0.0980 - 0.995i)T^{2} \) |
| 89 | \( 1 + (0.980 + 0.195i)T^{2} \) |
| 97 | \( 1 + (0.707 + 0.707i)T^{2} \) |
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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
−9.632383765150652051964745191912, −9.033928855005327442564150563189, −8.184603161840023964815383154421, −7.00970757711217610131590695389, −6.69494432368820500671665914249, −5.96174492544780660713238828984, −5.01249558475049120049446949798, −3.91890162886588906355905593084, −3.28647646727335537603935918508, −1.39707393629494513039505177592,
0.73778193474072524286579131944, 2.25982378835590047994265147658, 3.20249703442799407465133187577, 3.97765510597344670230435339605, 4.87290679788032821305011411818, 6.07981262843540501841544541105, 6.62496103902310166294797842204, 7.998637749172163710999126525758, 8.627309757263577889007850171504, 9.224268383021230360573622234080