L(s) = 1 | − 2.39·2-s − 3-s + 3.72·4-s − 2.51·5-s + 2.39·6-s − 4.13·8-s + 9-s + 6.02·10-s + 5.25·11-s − 3.72·12-s − 0.497·13-s + 2.51·15-s + 2.44·16-s + 7.20·17-s − 2.39·18-s + 5.61·19-s − 9.38·20-s − 12.5·22-s − 4.64·23-s + 4.13·24-s + 1.34·25-s + 1.19·26-s − 27-s + 2.78·29-s − 6.02·30-s − 5.51·31-s + 2.42·32-s + ⋯ |
L(s) = 1 | − 1.69·2-s − 0.577·3-s + 1.86·4-s − 1.12·5-s + 0.977·6-s − 1.46·8-s + 0.333·9-s + 1.90·10-s + 1.58·11-s − 1.07·12-s − 0.137·13-s + 0.650·15-s + 0.611·16-s + 1.74·17-s − 0.564·18-s + 1.28·19-s − 2.09·20-s − 2.68·22-s − 0.967·23-s + 0.844·24-s + 0.268·25-s + 0.233·26-s − 0.192·27-s + 0.518·29-s − 1.10·30-s − 0.990·31-s + 0.428·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 41 | \( 1 - T \) |
good | 2 | \( 1 + 2.39T + 2T^{2} \) |
| 5 | \( 1 + 2.51T + 5T^{2} \) |
| 11 | \( 1 - 5.25T + 11T^{2} \) |
| 13 | \( 1 + 0.497T + 13T^{2} \) |
| 17 | \( 1 - 7.20T + 17T^{2} \) |
| 19 | \( 1 - 5.61T + 19T^{2} \) |
| 23 | \( 1 + 4.64T + 23T^{2} \) |
| 29 | \( 1 - 2.78T + 29T^{2} \) |
| 31 | \( 1 + 5.51T + 31T^{2} \) |
| 37 | \( 1 + 9.17T + 37T^{2} \) |
| 43 | \( 1 + 11.4T + 43T^{2} \) |
| 47 | \( 1 - 8.72T + 47T^{2} \) |
| 53 | \( 1 + 10.5T + 53T^{2} \) |
| 59 | \( 1 - 1.20T + 59T^{2} \) |
| 61 | \( 1 + 9.52T + 61T^{2} \) |
| 67 | \( 1 + 4.94T + 67T^{2} \) |
| 71 | \( 1 + 5.28T + 71T^{2} \) |
| 73 | \( 1 + 0.0721T + 73T^{2} \) |
| 79 | \( 1 - 7.85T + 79T^{2} \) |
| 83 | \( 1 - 9.86T + 83T^{2} \) |
| 89 | \( 1 - 15.8T + 89T^{2} \) |
| 97 | \( 1 - 5.21T + 97T^{2} \) |
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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
−7.74415390954820429761897046018, −7.32057944388277064185121801122, −6.63500552530737227647470870821, −5.86291452008506859192770243183, −4.89853203718390066236179052790, −3.79069735440416142096734968067, −3.27651904085541748209327891295, −1.69175620136474834163604787866, −1.06607908178000666486128479813, 0,
1.06607908178000666486128479813, 1.69175620136474834163604787866, 3.27651904085541748209327891295, 3.79069735440416142096734968067, 4.89853203718390066236179052790, 5.86291452008506859192770243183, 6.63500552530737227647470870821, 7.32057944388277064185121801122, 7.74415390954820429761897046018