L(s) = 1 | + (0.0922 + 0.995i)2-s + (−0.243 + 0.489i)3-s + (−0.982 + 0.183i)4-s + (−0.510 − 0.197i)6-s + (−0.273 − 0.961i)8-s + (0.422 + 0.558i)9-s + (−1.83 + 0.342i)11-s + (0.149 − 0.526i)12-s + (0.932 − 0.361i)16-s + (−0.510 + 1.79i)17-s + (−0.517 + 0.471i)18-s + (−0.404 + 0.368i)19-s + (−0.510 − 1.79i)22-s + (0.538 + 0.100i)24-s + (−0.982 − 0.183i)25-s + ⋯ |
L(s) = 1 | + (0.0922 + 0.995i)2-s + (−0.243 + 0.489i)3-s + (−0.982 + 0.183i)4-s + (−0.510 − 0.197i)6-s + (−0.273 − 0.961i)8-s + (0.422 + 0.558i)9-s + (−1.83 + 0.342i)11-s + (0.149 − 0.526i)12-s + (0.932 − 0.361i)16-s + (−0.510 + 1.79i)17-s + (−0.517 + 0.471i)18-s + (−0.404 + 0.368i)19-s + (−0.510 − 1.79i)22-s + (0.538 + 0.100i)24-s + (−0.982 − 0.183i)25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1096 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.981 + 0.190i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1096 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.981 + 0.190i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6219898137\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6219898137\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.0922 - 0.995i)T \) |
| 137 | \( 1 + (-0.739 - 0.673i)T \) |
good | 3 | \( 1 + (0.243 - 0.489i)T + (-0.602 - 0.798i)T^{2} \) |
| 5 | \( 1 + (0.982 + 0.183i)T^{2} \) |
| 7 | \( 1 + (-0.445 - 0.895i)T^{2} \) |
| 11 | \( 1 + (1.83 - 0.342i)T + (0.932 - 0.361i)T^{2} \) |
| 13 | \( 1 + (-0.445 - 0.895i)T^{2} \) |
| 17 | \( 1 + (0.510 - 1.79i)T + (-0.850 - 0.526i)T^{2} \) |
| 19 | \( 1 + (0.404 - 0.368i)T + (0.0922 - 0.995i)T^{2} \) |
| 23 | \( 1 + (-0.739 + 0.673i)T^{2} \) |
| 29 | \( 1 + (-0.739 + 0.673i)T^{2} \) |
| 31 | \( 1 + (-0.0922 - 0.995i)T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - 0.891T + T^{2} \) |
| 43 | \( 1 + (-1.37 + 1.25i)T + (0.0922 - 0.995i)T^{2} \) |
| 47 | \( 1 + (0.273 - 0.961i)T^{2} \) |
| 53 | \( 1 + (-0.0922 + 0.995i)T^{2} \) |
| 59 | \( 1 + (0.111 + 0.147i)T + (-0.273 + 0.961i)T^{2} \) |
| 61 | \( 1 + (0.273 + 0.961i)T^{2} \) |
| 67 | \( 1 + (-0.465 - 0.288i)T + (0.445 + 0.895i)T^{2} \) |
| 71 | \( 1 + (-0.932 - 0.361i)T^{2} \) |
| 73 | \( 1 + (0.757 - 0.469i)T + (0.445 - 0.895i)T^{2} \) |
| 79 | \( 1 + (0.602 - 0.798i)T^{2} \) |
| 83 | \( 1 + (-0.465 - 1.63i)T + (-0.850 + 0.526i)T^{2} \) |
| 89 | \( 1 + (-0.136 + 1.47i)T + (-0.982 - 0.183i)T^{2} \) |
| 97 | \( 1 + (-1.93 + 0.361i)T + (0.932 - 0.361i)T^{2} \) |
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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
−10.44744723565074305932678722362, −9.723230256829359955534129384569, −8.619634317778095216113934509015, −7.891351467530308836889471801344, −7.35527890102175764572832946666, −6.06972902632981850502715064939, −5.51210352110402745537187298085, −4.54466629359101272526988183238, −3.87722718011056479422080242172, −2.22094079280064642470444996598,
0.54948366249340831418208994508, 2.22030226116450901912061679435, 3.02722227787454312454111571968, 4.31921556809610453106409284069, 5.18970591340642841394114901630, 6.04637779487629424030776490699, 7.29960072532134448465139636942, 7.952814262067867515870233732002, 9.082677219272717405184416373106, 9.690015671218948904999924217421