L(s) = 1 | + 2-s − 0.431·3-s + 4-s + 0.638·5-s − 0.431·6-s + 2.88·7-s + 8-s − 2.81·9-s + 0.638·10-s − 4.60·11-s − 0.431·12-s + 2.88·14-s − 0.275·15-s + 16-s + 6.56·17-s − 2.81·18-s + 19-s + 0.638·20-s − 1.24·21-s − 4.60·22-s + 8.91·23-s − 0.431·24-s − 4.59·25-s + 2.50·27-s + 2.88·28-s + 0.265·29-s − 0.275·30-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.248·3-s + 0.5·4-s + 0.285·5-s − 0.175·6-s + 1.09·7-s + 0.353·8-s − 0.938·9-s + 0.201·10-s − 1.38·11-s − 0.124·12-s + 0.771·14-s − 0.0710·15-s + 0.250·16-s + 1.59·17-s − 0.663·18-s + 0.229·19-s + 0.142·20-s − 0.271·21-s − 0.981·22-s + 1.85·23-s − 0.0879·24-s − 0.918·25-s + 0.482·27-s + 0.545·28-s + 0.0493·29-s − 0.0502·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.285162559\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.285162559\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 13 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 3 | \( 1 + 0.431T + 3T^{2} \) |
| 5 | \( 1 - 0.638T + 5T^{2} \) |
| 7 | \( 1 - 2.88T + 7T^{2} \) |
| 11 | \( 1 + 4.60T + 11T^{2} \) |
| 17 | \( 1 - 6.56T + 17T^{2} \) |
| 23 | \( 1 - 8.91T + 23T^{2} \) |
| 29 | \( 1 - 0.265T + 29T^{2} \) |
| 31 | \( 1 + 2.61T + 31T^{2} \) |
| 37 | \( 1 - 5.69T + 37T^{2} \) |
| 41 | \( 1 - 0.820T + 41T^{2} \) |
| 43 | \( 1 - 0.882T + 43T^{2} \) |
| 47 | \( 1 + 1.44T + 47T^{2} \) |
| 53 | \( 1 + 8.64T + 53T^{2} \) |
| 59 | \( 1 + 8.55T + 59T^{2} \) |
| 61 | \( 1 - 6.21T + 61T^{2} \) |
| 67 | \( 1 - 10.0T + 67T^{2} \) |
| 71 | \( 1 - 3.88T + 71T^{2} \) |
| 73 | \( 1 - 8.05T + 73T^{2} \) |
| 79 | \( 1 + 9.12T + 79T^{2} \) |
| 83 | \( 1 - 8.35T + 83T^{2} \) |
| 89 | \( 1 - 15.1T + 89T^{2} \) |
| 97 | \( 1 - 2.31T + 97T^{2} \) |
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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
−7.77760807221221582620243429358, −7.54461147913122580573286176915, −6.37012999679573397005006700928, −5.64446900496037068512556343889, −5.16103383036464072960762571258, −4.78430866646332543567082013420, −3.48457885128111647317672453411, −2.86879123812297882393324160382, −2.00294163812052367702028365700, −0.865085936445304537548457529125,
0.865085936445304537548457529125, 2.00294163812052367702028365700, 2.86879123812297882393324160382, 3.48457885128111647317672453411, 4.78430866646332543567082013420, 5.16103383036464072960762571258, 5.64446900496037068512556343889, 6.37012999679573397005006700928, 7.54461147913122580573286176915, 7.77760807221221582620243429358