L(s) = 1 | + 0.580·3-s − 0.315·7-s − 2.66·9-s + 4.34·11-s + 4.58·13-s + 0.917·17-s − 2.76·19-s − 0.182·21-s + 23-s − 3.28·27-s + 7.03·29-s + 0.867·31-s + 2.52·33-s + 4.68·37-s + 2.65·39-s + 4.69·41-s − 9.08·43-s − 8.24·47-s − 6.90·49-s + 0.532·51-s + 10.9·53-s − 1.60·57-s − 1.50·59-s + 11.4·61-s + 0.839·63-s + 1.68·67-s + 0.580·69-s + ⋯ |
L(s) = 1 | + 0.335·3-s − 0.119·7-s − 0.887·9-s + 1.31·11-s + 1.27·13-s + 0.222·17-s − 0.634·19-s − 0.0399·21-s + 0.208·23-s − 0.632·27-s + 1.30·29-s + 0.155·31-s + 0.439·33-s + 0.770·37-s + 0.425·39-s + 0.732·41-s − 1.38·43-s − 1.20·47-s − 0.985·49-s + 0.0745·51-s + 1.49·53-s − 0.212·57-s − 0.195·59-s + 1.46·61-s + 0.105·63-s + 0.205·67-s + 0.0698·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.528667219\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.528667219\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 - T \) |
good | 3 | \( 1 - 0.580T + 3T^{2} \) |
| 7 | \( 1 + 0.315T + 7T^{2} \) |
| 11 | \( 1 - 4.34T + 11T^{2} \) |
| 13 | \( 1 - 4.58T + 13T^{2} \) |
| 17 | \( 1 - 0.917T + 17T^{2} \) |
| 19 | \( 1 + 2.76T + 19T^{2} \) |
| 29 | \( 1 - 7.03T + 29T^{2} \) |
| 31 | \( 1 - 0.867T + 31T^{2} \) |
| 37 | \( 1 - 4.68T + 37T^{2} \) |
| 41 | \( 1 - 4.69T + 41T^{2} \) |
| 43 | \( 1 + 9.08T + 43T^{2} \) |
| 47 | \( 1 + 8.24T + 47T^{2} \) |
| 53 | \( 1 - 10.9T + 53T^{2} \) |
| 59 | \( 1 + 1.50T + 59T^{2} \) |
| 61 | \( 1 - 11.4T + 61T^{2} \) |
| 67 | \( 1 - 1.68T + 67T^{2} \) |
| 71 | \( 1 - 5.36T + 71T^{2} \) |
| 73 | \( 1 + 10.1T + 73T^{2} \) |
| 79 | \( 1 + 7.39T + 79T^{2} \) |
| 83 | \( 1 - 15.3T + 83T^{2} \) |
| 89 | \( 1 + 10.9T + 89T^{2} \) |
| 97 | \( 1 - 14.2T + 97T^{2} \) |
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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
−7.939894674861252375399113683389, −6.82841645597574895871970905534, −6.39533969083967319623221365323, −5.84832692070584405845400539836, −4.91479088223020218074915070237, −4.04959400912490096033949677735, −3.47551405357879253207830891569, −2.73222012387525731846152339378, −1.69740170914341676064329585847, −0.78139247131479462055369108162,
0.78139247131479462055369108162, 1.69740170914341676064329585847, 2.73222012387525731846152339378, 3.47551405357879253207830891569, 4.04959400912490096033949677735, 4.91479088223020218074915070237, 5.84832692070584405845400539836, 6.39533969083967319623221365323, 6.82841645597574895871970905534, 7.939894674861252375399113683389