L(s) = 1 | + 1.67·3-s + 3.57·7-s − 0.177·9-s + 2.43·11-s + 4.65·13-s + 6.59·17-s + 3.50·19-s + 6.00·21-s + 4.36·23-s − 5.33·27-s + 3.64·29-s + 2.93·31-s + 4.08·33-s − 5.33·37-s + 7.82·39-s − 0.295·41-s − 6.57·43-s − 7.34·47-s + 5.79·49-s + 11.0·51-s − 0.486·53-s + 5.88·57-s + 8.80·59-s + 9.00·61-s − 0.636·63-s − 10.0·67-s + 7.33·69-s + ⋯ |
L(s) = 1 | + 0.969·3-s + 1.35·7-s − 0.0593·9-s + 0.732·11-s + 1.29·13-s + 1.59·17-s + 0.804·19-s + 1.31·21-s + 0.910·23-s − 1.02·27-s + 0.677·29-s + 0.526·31-s + 0.710·33-s − 0.876·37-s + 1.25·39-s − 0.0462·41-s − 1.00·43-s − 1.07·47-s + 0.828·49-s + 1.55·51-s − 0.0668·53-s + 0.779·57-s + 1.14·59-s + 1.15·61-s − 0.0802·63-s − 1.22·67-s + 0.883·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.643229788\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.643229788\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 - 1.67T + 3T^{2} \) |
| 7 | \( 1 - 3.57T + 7T^{2} \) |
| 11 | \( 1 - 2.43T + 11T^{2} \) |
| 13 | \( 1 - 4.65T + 13T^{2} \) |
| 17 | \( 1 - 6.59T + 17T^{2} \) |
| 19 | \( 1 - 3.50T + 19T^{2} \) |
| 23 | \( 1 - 4.36T + 23T^{2} \) |
| 29 | \( 1 - 3.64T + 29T^{2} \) |
| 31 | \( 1 - 2.93T + 31T^{2} \) |
| 37 | \( 1 + 5.33T + 37T^{2} \) |
| 41 | \( 1 + 0.295T + 41T^{2} \) |
| 43 | \( 1 + 6.57T + 43T^{2} \) |
| 47 | \( 1 + 7.34T + 47T^{2} \) |
| 53 | \( 1 + 0.486T + 53T^{2} \) |
| 59 | \( 1 - 8.80T + 59T^{2} \) |
| 61 | \( 1 - 9.00T + 61T^{2} \) |
| 67 | \( 1 + 10.0T + 67T^{2} \) |
| 71 | \( 1 + 10.7T + 71T^{2} \) |
| 73 | \( 1 - 15.3T + 73T^{2} \) |
| 79 | \( 1 - 7.53T + 79T^{2} \) |
| 83 | \( 1 - 1.56T + 83T^{2} \) |
| 89 | \( 1 + 17.0T + 89T^{2} \) |
| 97 | \( 1 + 14.9T + 97T^{2} \) |
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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
−7.892980912920484005933767644307, −7.13513437844808925482330048280, −6.35292293308696356026593170158, −5.43376284786903114541579400193, −5.01565354390551302508907590772, −3.93017431620062842154885738604, −3.42872220764277888711019464267, −2.71300452347304465814777103448, −1.52044654261762649398528719080, −1.16501712302622099934802015926,
1.16501712302622099934802015926, 1.52044654261762649398528719080, 2.71300452347304465814777103448, 3.42872220764277888711019464267, 3.93017431620062842154885738604, 5.01565354390551302508907590772, 5.43376284786903114541579400193, 6.35292293308696356026593170158, 7.13513437844808925482330048280, 7.892980912920484005933767644307