L(s) = 1 | + 1.07i·3-s − 11.6i·5-s + 2.67·7-s + 25.8·9-s + 63.9i·11-s + 50.0i·13-s + 12.4·15-s + 72.4·17-s − 27.4i·19-s + 2.85i·21-s − 139.·23-s − 10.3·25-s + 56.5i·27-s − 93.3i·29-s − 188.·31-s + ⋯ |
L(s) = 1 | + 0.205i·3-s − 1.04i·5-s + 0.144·7-s + 0.957·9-s + 1.75i·11-s + 1.06i·13-s + 0.214·15-s + 1.03·17-s − 0.332i·19-s + 0.0297i·21-s − 1.26·23-s − 0.0826·25-s + 0.403i·27-s − 0.598i·29-s − 1.09·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.727766883\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.727766883\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
good | 3 | \( 1 - 1.07iT - 27T^{2} \) |
| 5 | \( 1 + 11.6iT - 125T^{2} \) |
| 7 | \( 1 - 2.67T + 343T^{2} \) |
| 11 | \( 1 - 63.9iT - 1.33e3T^{2} \) |
| 13 | \( 1 - 50.0iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 72.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 27.4iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 139.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 93.3iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 188.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 118. iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 104.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 44.5iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 488.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 211. iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 402. iT - 2.05e5T^{2} \) |
| 61 | \( 1 - 322. iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 196. iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 453.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 259.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 323.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 797. iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 866.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 936.T + 9.12e5T^{2} \) |
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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
−9.714461910524542795523078845721, −9.173130696438043461749094485874, −8.054171950078921126514166541377, −7.33992164047772251726021385976, −6.47593285981272325196326229059, −5.14048617427916447225367808129, −4.57068707171562021871722964467, −3.84802929255001388143033960710, −2.03949103415622623766217241381, −1.31235818781141497303487840830,
0.43573069393050493713773799589, 1.74733608780499775319022653878, 3.21024796772076960535820625180, 3.60219565301878950634830348388, 5.19899655542969075539398622359, 6.01964212813464593607904323934, 6.78490782853473769522827544620, 7.82197590558746291025758518326, 8.206892430213144372373535734034, 9.519975768623303627175592306825