L(s) = 1 | − 3-s + 5-s + (−1.59 + 2.76i)7-s + 9-s + (2.11 − 3.66i)11-s + (1.09 + 1.88i)13-s − 15-s + (−2.56 − 4.43i)17-s + (−0.325 − 0.564i)19-s + (1.59 − 2.76i)21-s + (4.02 + 6.96i)23-s + 25-s − 27-s + (−3.94 + 6.83i)29-s + (4.23 − 7.34i)31-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.447·5-s + (−0.602 + 1.04i)7-s + 0.333·9-s + (0.638 − 1.10i)11-s + (0.302 + 0.523i)13-s − 0.258·15-s + (−0.621 − 1.07i)17-s + (−0.0747 − 0.129i)19-s + (0.347 − 0.602i)21-s + (0.838 + 1.45i)23-s + 0.200·25-s − 0.192·27-s + (−0.733 + 1.27i)29-s + (0.761 − 1.31i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.921 - 0.387i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.921 - 0.387i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.562949375\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.562949375\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 67 | \( 1 + (-5.59 + 5.97i)T \) |
good | 7 | \( 1 + (1.59 - 2.76i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.11 + 3.66i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.09 - 1.88i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (2.56 + 4.43i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.325 + 0.564i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-4.02 - 6.96i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (3.94 - 6.83i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-4.23 + 7.34i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (1.75 + 3.03i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.135 + 0.234i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 - 8.32T + 43T^{2} \) |
| 47 | \( 1 + (0.171 - 0.297i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 7.89T + 53T^{2} \) |
| 59 | \( 1 + 4.39T + 59T^{2} \) |
| 61 | \( 1 + (7.05 + 12.2i)T + (-30.5 + 52.8i)T^{2} \) |
| 71 | \( 1 + (-1.45 + 2.51i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-3.18 - 5.51i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.598 + 1.03i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-0.963 - 1.66i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 3.23T + 89T^{2} \) |
| 97 | \( 1 + (-5.58 - 9.67i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−8.834918720449767846210195869349, −7.65278292555379184987498264851, −6.81155515879638511274564564770, −6.20445648894534011015953716717, −5.62290629488595472155805944514, −4.97941165329859241402556608251, −3.82657633013203640532752956288, −3.01067192267472527827193574935, −2.03226374180278064634066167084, −0.798963982856257673985654999283,
0.70308317341682448871715088178, 1.73594354325011122156613726747, 2.89313945188613768049542772971, 4.12915228874347625224774600124, 4.38526242637266340904440570214, 5.51550609275166861628065317202, 6.38207965980780403112369416568, 6.75716799671788973692770937819, 7.45509242838139544429537272385, 8.448993553663534082547952844653