L(s) = 1 | + 3-s − 1.31·7-s + 9-s + 1.55·11-s + 2.42·13-s − 1.31·17-s − 1.02·19-s − 1.31·21-s + 5.00·23-s + 27-s − 5.90·29-s + 8.85·31-s + 1.55·33-s + 5.02·37-s + 2.42·39-s + 7.95·41-s − 6.84·43-s + 7.75·47-s − 5.26·49-s − 1.31·51-s − 3.95·53-s − 1.02·57-s − 7.85·59-s − 3.50·61-s − 1.31·63-s − 8.33·67-s + 5.00·69-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.498·7-s + 0.333·9-s + 0.468·11-s + 0.673·13-s − 0.318·17-s − 0.234·19-s − 0.287·21-s + 1.04·23-s + 0.192·27-s − 1.09·29-s + 1.58·31-s + 0.270·33-s + 0.826·37-s + 0.388·39-s + 1.24·41-s − 1.04·43-s + 1.13·47-s − 0.751·49-s − 0.184·51-s − 0.542·53-s − 0.135·57-s − 1.02·59-s − 0.448·61-s − 0.166·63-s − 1.01·67-s + 0.602·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7500 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.595776524\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.595776524\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 1.31T + 7T^{2} \) |
| 11 | \( 1 - 1.55T + 11T^{2} \) |
| 13 | \( 1 - 2.42T + 13T^{2} \) |
| 17 | \( 1 + 1.31T + 17T^{2} \) |
| 19 | \( 1 + 1.02T + 19T^{2} \) |
| 23 | \( 1 - 5.00T + 23T^{2} \) |
| 29 | \( 1 + 5.90T + 29T^{2} \) |
| 31 | \( 1 - 8.85T + 31T^{2} \) |
| 37 | \( 1 - 5.02T + 37T^{2} \) |
| 41 | \( 1 - 7.95T + 41T^{2} \) |
| 43 | \( 1 + 6.84T + 43T^{2} \) |
| 47 | \( 1 - 7.75T + 47T^{2} \) |
| 53 | \( 1 + 3.95T + 53T^{2} \) |
| 59 | \( 1 + 7.85T + 59T^{2} \) |
| 61 | \( 1 + 3.50T + 61T^{2} \) |
| 67 | \( 1 + 8.33T + 67T^{2} \) |
| 71 | \( 1 + 12.9T + 71T^{2} \) |
| 73 | \( 1 - 12.7T + 73T^{2} \) |
| 79 | \( 1 + 1.25T + 79T^{2} \) |
| 83 | \( 1 - 8.20T + 83T^{2} \) |
| 89 | \( 1 - 9.57T + 89T^{2} \) |
| 97 | \( 1 - 16.2T + 97T^{2} \) |
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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
−7.83951782852614459402085143156, −7.30823870271713427358118048674, −6.34052914492721883279625574463, −6.12335925142544839542143632503, −4.90721300839311553213613202288, −4.26706572297873901333877016238, −3.43673073379048270557322640901, −2.82991165368917523497163775648, −1.82697005222597703442515736068, −0.803090753023393581601096676886,
0.803090753023393581601096676886, 1.82697005222597703442515736068, 2.82991165368917523497163775648, 3.43673073379048270557322640901, 4.26706572297873901333877016238, 4.90721300839311553213613202288, 6.12335925142544839542143632503, 6.34052914492721883279625574463, 7.30823870271713427358118048674, 7.83951782852614459402085143156