L(s) = 1 | − 0.284·3-s + 1.91·5-s + 0.801·7-s − 2.91·9-s − 3.18·11-s + 6.28·13-s − 0.544·15-s − 2.22·17-s − 19-s − 0.228·21-s + 2.43·23-s − 1.33·25-s + 1.68·27-s + 5.69·29-s + 7.04·31-s + 0.907·33-s + 1.53·35-s − 1.46·37-s − 1.78·39-s − 5.70·41-s + 0.714·43-s − 5.58·45-s + 2.67·47-s − 6.35·49-s + 0.634·51-s − 53-s − 6.10·55-s + ⋯ |
L(s) = 1 | − 0.164·3-s + 0.855·5-s + 0.303·7-s − 0.972·9-s − 0.961·11-s + 1.74·13-s − 0.140·15-s − 0.540·17-s − 0.229·19-s − 0.0498·21-s + 0.507·23-s − 0.267·25-s + 0.324·27-s + 1.05·29-s + 1.26·31-s + 0.158·33-s + 0.259·35-s − 0.240·37-s − 0.286·39-s − 0.891·41-s + 0.108·43-s − 0.832·45-s + 0.390·47-s − 0.908·49-s + 0.0888·51-s − 0.137·53-s − 0.822·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4028 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.032470050\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.032470050\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 + T \) |
| 53 | \( 1 + T \) |
good | 3 | \( 1 + 0.284T + 3T^{2} \) |
| 5 | \( 1 - 1.91T + 5T^{2} \) |
| 7 | \( 1 - 0.801T + 7T^{2} \) |
| 11 | \( 1 + 3.18T + 11T^{2} \) |
| 13 | \( 1 - 6.28T + 13T^{2} \) |
| 17 | \( 1 + 2.22T + 17T^{2} \) |
| 23 | \( 1 - 2.43T + 23T^{2} \) |
| 29 | \( 1 - 5.69T + 29T^{2} \) |
| 31 | \( 1 - 7.04T + 31T^{2} \) |
| 37 | \( 1 + 1.46T + 37T^{2} \) |
| 41 | \( 1 + 5.70T + 41T^{2} \) |
| 43 | \( 1 - 0.714T + 43T^{2} \) |
| 47 | \( 1 - 2.67T + 47T^{2} \) |
| 59 | \( 1 - 13.9T + 59T^{2} \) |
| 61 | \( 1 + 4.47T + 61T^{2} \) |
| 67 | \( 1 - 13.8T + 67T^{2} \) |
| 71 | \( 1 - 10.0T + 71T^{2} \) |
| 73 | \( 1 - 9.48T + 73T^{2} \) |
| 79 | \( 1 - 11.4T + 79T^{2} \) |
| 83 | \( 1 + 9.21T + 83T^{2} \) |
| 89 | \( 1 + 0.388T + 89T^{2} \) |
| 97 | \( 1 - 15.9T + 97T^{2} \) |
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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
−8.375738842488283432383599305178, −8.036719675727630556037195189664, −6.62767391300877584006711370646, −6.31140103130639694570198202496, −5.44493483821881030123742745679, −4.94064603137991035588000751567, −3.77865114848742731974832639086, −2.83664902158085905631636227035, −2.03888871560824691352791862667, −0.827881948890706335801164168670,
0.827881948890706335801164168670, 2.03888871560824691352791862667, 2.83664902158085905631636227035, 3.77865114848742731974832639086, 4.94064603137991035588000751567, 5.44493483821881030123742745679, 6.31140103130639694570198202496, 6.62767391300877584006711370646, 8.036719675727630556037195189664, 8.375738842488283432383599305178