L(s) = 1 | − 2-s + 0.335·3-s + 4-s + 5-s − 0.335·6-s − 3.89·7-s − 8-s − 2.88·9-s − 10-s − 11-s + 0.335·12-s + 1.74·13-s + 3.89·14-s + 0.335·15-s + 16-s + 0.959·17-s + 2.88·18-s − 0.980·19-s + 20-s − 1.30·21-s + 22-s + 6.94·23-s − 0.335·24-s + 25-s − 1.74·26-s − 1.97·27-s − 3.89·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.193·3-s + 0.5·4-s + 0.447·5-s − 0.137·6-s − 1.47·7-s − 0.353·8-s − 0.962·9-s − 0.316·10-s − 0.301·11-s + 0.0969·12-s + 0.484·13-s + 1.04·14-s + 0.0867·15-s + 0.250·16-s + 0.232·17-s + 0.680·18-s − 0.224·19-s + 0.223·20-s − 0.285·21-s + 0.213·22-s + 1.44·23-s − 0.0685·24-s + 0.200·25-s − 0.342·26-s − 0.380·27-s − 0.736·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 73 | \( 1 - T \) |
good | 3 | \( 1 - 0.335T + 3T^{2} \) |
| 7 | \( 1 + 3.89T + 7T^{2} \) |
| 13 | \( 1 - 1.74T + 13T^{2} \) |
| 17 | \( 1 - 0.959T + 17T^{2} \) |
| 19 | \( 1 + 0.980T + 19T^{2} \) |
| 23 | \( 1 - 6.94T + 23T^{2} \) |
| 29 | \( 1 + 4.84T + 29T^{2} \) |
| 31 | \( 1 - 5.52T + 31T^{2} \) |
| 37 | \( 1 - 11.5T + 37T^{2} \) |
| 41 | \( 1 - 3.60T + 41T^{2} \) |
| 43 | \( 1 + 10.3T + 43T^{2} \) |
| 47 | \( 1 + 0.509T + 47T^{2} \) |
| 53 | \( 1 + 13.9T + 53T^{2} \) |
| 59 | \( 1 - 8.18T + 59T^{2} \) |
| 61 | \( 1 + 1.07T + 61T^{2} \) |
| 67 | \( 1 + 5.45T + 67T^{2} \) |
| 71 | \( 1 + 1.51T + 71T^{2} \) |
| 79 | \( 1 - 1.12T + 79T^{2} \) |
| 83 | \( 1 - 6.23T + 83T^{2} \) |
| 89 | \( 1 + 4.85T + 89T^{2} \) |
| 97 | \( 1 + 11.4T + 97T^{2} \) |
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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
−7.57063631790780722631021596239, −6.71881776495209868961910598642, −6.21716047811459154189449847602, −5.70395086005260243458417585631, −4.75275689041625063775422562991, −3.50228971476183962734292657726, −3.00719404703916640093969032845, −2.35316270191351374435935614608, −1.07706981266973021538952561315, 0,
1.07706981266973021538952561315, 2.35316270191351374435935614608, 3.00719404703916640093969032845, 3.50228971476183962734292657726, 4.75275689041625063775422562991, 5.70395086005260243458417585631, 6.21716047811459154189449847602, 6.71881776495209868961910598642, 7.57063631790780722631021596239