| L(s) = 1 | − 2.44·2-s + 3.97·4-s − 2.87·5-s − 1.56·7-s − 4.82·8-s + 7.02·10-s + 6.34·11-s + 0.803·13-s + 3.81·14-s + 3.83·16-s + 3.25·19-s − 11.4·20-s − 15.5·22-s + 6.17·23-s + 3.25·25-s − 1.96·26-s − 6.20·28-s + 1.52·29-s + 2.17·31-s + 0.263·32-s + 4.48·35-s + 0.636·37-s − 7.94·38-s + 13.8·40-s − 5.61·41-s − 11.0·43-s + 25.1·44-s + ⋯ |
| L(s) = 1 | − 1.72·2-s + 1.98·4-s − 1.28·5-s − 0.590·7-s − 1.70·8-s + 2.22·10-s + 1.91·11-s + 0.222·13-s + 1.02·14-s + 0.959·16-s + 0.745·19-s − 2.55·20-s − 3.30·22-s + 1.28·23-s + 0.651·25-s − 0.385·26-s − 1.17·28-s + 0.283·29-s + 0.390·31-s + 0.0465·32-s + 0.758·35-s + 0.104·37-s − 1.28·38-s + 2.19·40-s − 0.877·41-s − 1.68·43-s + 3.79·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5810164235\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5810164235\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 + 2.44T + 2T^{2} \) |
| 5 | \( 1 + 2.87T + 5T^{2} \) |
| 7 | \( 1 + 1.56T + 7T^{2} \) |
| 11 | \( 1 - 6.34T + 11T^{2} \) |
| 13 | \( 1 - 0.803T + 13T^{2} \) |
| 19 | \( 1 - 3.25T + 19T^{2} \) |
| 23 | \( 1 - 6.17T + 23T^{2} \) |
| 29 | \( 1 - 1.52T + 29T^{2} \) |
| 31 | \( 1 - 2.17T + 31T^{2} \) |
| 37 | \( 1 - 0.636T + 37T^{2} \) |
| 41 | \( 1 + 5.61T + 41T^{2} \) |
| 43 | \( 1 + 11.0T + 43T^{2} \) |
| 47 | \( 1 + 7.46T + 47T^{2} \) |
| 53 | \( 1 + 1.34T + 53T^{2} \) |
| 59 | \( 1 + 4.03T + 59T^{2} \) |
| 61 | \( 1 - 8.36T + 61T^{2} \) |
| 67 | \( 1 - 7.75T + 67T^{2} \) |
| 71 | \( 1 - 4.51T + 71T^{2} \) |
| 73 | \( 1 - 8.28T + 73T^{2} \) |
| 79 | \( 1 - 1.50T + 79T^{2} \) |
| 83 | \( 1 + 7.52T + 83T^{2} \) |
| 89 | \( 1 - 6.12T + 89T^{2} \) |
| 97 | \( 1 - 4.84T + 97T^{2} \) |
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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
−8.811098878355366899326422436877, −8.311996803123735903061593222406, −7.53041012564089635351910644477, −6.66939221861003876725986657644, −6.55174206883204988399295758724, −4.90937712238660022675455007569, −3.73802778952328153197774722326, −3.13364398619885055508889554251, −1.57172453538425916947303360436, −0.66006323600139835036125663189,
0.66006323600139835036125663189, 1.57172453538425916947303360436, 3.13364398619885055508889554251, 3.73802778952328153197774722326, 4.90937712238660022675455007569, 6.55174206883204988399295758724, 6.66939221861003876725986657644, 7.53041012564089635351910644477, 8.311996803123735903061593222406, 8.811098878355366899326422436877
