L(s) = 1 | + 0.843·2-s − 3-s − 1.28·4-s + 5-s − 0.843·6-s − 1.98·7-s − 2.77·8-s + 9-s + 0.843·10-s + 2.08·11-s + 1.28·12-s − 0.684·13-s − 1.67·14-s − 15-s + 0.233·16-s + 4.82·17-s + 0.843·18-s − 3.22·19-s − 1.28·20-s + 1.98·21-s + 1.76·22-s − 2.25·23-s + 2.77·24-s + 25-s − 0.577·26-s − 27-s + 2.56·28-s + ⋯ |
L(s) = 1 | + 0.596·2-s − 0.577·3-s − 0.643·4-s + 0.447·5-s − 0.344·6-s − 0.751·7-s − 0.981·8-s + 0.333·9-s + 0.266·10-s + 0.629·11-s + 0.371·12-s − 0.189·13-s − 0.448·14-s − 0.258·15-s + 0.0583·16-s + 1.17·17-s + 0.198·18-s − 0.738·19-s − 0.287·20-s + 0.433·21-s + 0.375·22-s − 0.469·23-s + 0.566·24-s + 0.200·25-s − 0.113·26-s − 0.192·27-s + 0.483·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 401 | \( 1 - T \) |
good | 2 | \( 1 - 0.843T + 2T^{2} \) |
| 7 | \( 1 + 1.98T + 7T^{2} \) |
| 11 | \( 1 - 2.08T + 11T^{2} \) |
| 13 | \( 1 + 0.684T + 13T^{2} \) |
| 17 | \( 1 - 4.82T + 17T^{2} \) |
| 19 | \( 1 + 3.22T + 19T^{2} \) |
| 23 | \( 1 + 2.25T + 23T^{2} \) |
| 29 | \( 1 + 0.651T + 29T^{2} \) |
| 31 | \( 1 + 1.09T + 31T^{2} \) |
| 37 | \( 1 + 6.34T + 37T^{2} \) |
| 41 | \( 1 - 6.70T + 41T^{2} \) |
| 43 | \( 1 + 1.40T + 43T^{2} \) |
| 47 | \( 1 - 7.23T + 47T^{2} \) |
| 53 | \( 1 - 4.84T + 53T^{2} \) |
| 59 | \( 1 - 0.173T + 59T^{2} \) |
| 61 | \( 1 + 12.5T + 61T^{2} \) |
| 67 | \( 1 - 8.21T + 67T^{2} \) |
| 71 | \( 1 - 5.11T + 71T^{2} \) |
| 73 | \( 1 + 2.02T + 73T^{2} \) |
| 79 | \( 1 - 13.3T + 79T^{2} \) |
| 83 | \( 1 - 4.15T + 83T^{2} \) |
| 89 | \( 1 + 4.45T + 89T^{2} \) |
| 97 | \( 1 + 6.46T + 97T^{2} \) |
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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
−7.63457406513707215060618956708, −6.71136048021505207851396910959, −6.14806366060328768220077151296, −5.58392665879699939479831342365, −4.92790536818000736602955940378, −4.01602603656850504715033382613, −3.51352175622727999954641471154, −2.47915645288322922559077987709, −1.18895313516457969368769796168, 0,
1.18895313516457969368769796168, 2.47915645288322922559077987709, 3.51352175622727999954641471154, 4.01602603656850504715033382613, 4.92790536818000736602955940378, 5.58392665879699939479831342365, 6.14806366060328768220077151296, 6.71136048021505207851396910959, 7.63457406513707215060618956708