L(s) = 1 | − 2.84·3-s + 0.245·5-s + 0.879·7-s + 5.08·9-s + 0.576·11-s − 2.27·13-s − 0.697·15-s + 0.418·17-s − 19-s − 2.50·21-s + 3.03·23-s − 4.93·25-s − 5.91·27-s + 6.53·29-s + 7.78·31-s − 1.63·33-s + 0.215·35-s + 1.99·37-s + 6.46·39-s − 9.28·41-s + 9.85·43-s + 1.24·45-s − 7.82·47-s − 6.22·49-s − 1.19·51-s − 53-s + 0.141·55-s + ⋯ |
L(s) = 1 | − 1.64·3-s + 0.109·5-s + 0.332·7-s + 1.69·9-s + 0.173·11-s − 0.630·13-s − 0.179·15-s + 0.101·17-s − 0.229·19-s − 0.545·21-s + 0.633·23-s − 0.987·25-s − 1.13·27-s + 1.21·29-s + 1.39·31-s − 0.285·33-s + 0.0364·35-s + 0.328·37-s + 1.03·39-s − 1.45·41-s + 1.50·43-s + 0.185·45-s − 1.14·47-s − 0.889·49-s − 0.166·51-s − 0.137·53-s + 0.0190·55-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
\(L(1)\) |
\(\approx\) |
\(0.9453163699\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9453163699\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.84T + 3T^{2} \) |
| 5 | \( 1 - 0.245T + 5T^{2} \) |
| 7 | \( 1 - 0.879T + 7T^{2} \) |
| 11 | \( 1 - 0.576T + 11T^{2} \) |
| 13 | \( 1 + 2.27T + 13T^{2} \) |
| 17 | \( 1 - 0.418T + 17T^{2} \) |
| 23 | \( 1 - 3.03T + 23T^{2} \) |
| 29 | \( 1 - 6.53T + 29T^{2} \) |
| 31 | \( 1 - 7.78T + 31T^{2} \) |
| 37 | \( 1 - 1.99T + 37T^{2} \) |
| 41 | \( 1 + 9.28T + 41T^{2} \) |
| 43 | \( 1 - 9.85T + 43T^{2} \) |
| 47 | \( 1 + 7.82T + 47T^{2} \) |
| 59 | \( 1 + 5.86T + 59T^{2} \) |
| 61 | \( 1 + 4.17T + 61T^{2} \) |
| 67 | \( 1 + 2.10T + 67T^{2} \) |
| 71 | \( 1 - 1.58T + 71T^{2} \) |
| 73 | \( 1 - 10.9T + 73T^{2} \) |
| 79 | \( 1 - 4.10T + 79T^{2} \) |
| 83 | \( 1 - 14.2T + 83T^{2} \) |
| 89 | \( 1 + 9.95T + 89T^{2} \) |
| 97 | \( 1 - 0.802T + 97T^{2} \) |
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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
−8.282310493357715777853699600021, −7.63219020880177300582932370755, −6.58834491132490930066624584562, −6.39081595480760740239764682020, −5.38732641098262300683021398751, −4.85345781557714550088323397429, −4.22223573944278962350221232151, −2.92433278523042326087411347243, −1.67396069628808312538920884594, −0.62263952655518816933729922097,
0.62263952655518816933729922097, 1.67396069628808312538920884594, 2.92433278523042326087411347243, 4.22223573944278962350221232151, 4.85345781557714550088323397429, 5.38732641098262300683021398751, 6.39081595480760740239764682020, 6.58834491132490930066624584562, 7.63219020880177300582932370755, 8.282310493357715777853699600021