| L(s) = 1 | + 4·5-s + 11.3i·7-s + 14.1i·11-s − 20·13-s + 10·17-s + 14.1i·19-s + 11.3i·23-s − 9·25-s + 20·29-s + 45.2i·35-s − 20·37-s − 30·41-s − 2.82i·43-s − 67.8i·47-s − 79.0·49-s + ⋯ |
| L(s) = 1 | + 0.800·5-s + 1.61i·7-s + 1.28i·11-s − 1.53·13-s + 0.588·17-s + 0.744i·19-s + 0.491i·23-s − 0.359·25-s + 0.689·29-s + 1.29i·35-s − 0.540·37-s − 0.731·41-s − 0.0657i·43-s − 1.44i·47-s − 1.61·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.152368407\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.152368407\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 4T + 25T^{2} \) |
| 7 | \( 1 - 11.3iT - 49T^{2} \) |
| 11 | \( 1 - 14.1iT - 121T^{2} \) |
| 13 | \( 1 + 20T + 169T^{2} \) |
| 17 | \( 1 - 10T + 289T^{2} \) |
| 19 | \( 1 - 14.1iT - 361T^{2} \) |
| 23 | \( 1 - 11.3iT - 529T^{2} \) |
| 29 | \( 1 - 20T + 841T^{2} \) |
| 31 | \( 1 - 961T^{2} \) |
| 37 | \( 1 + 20T + 1.36e3T^{2} \) |
| 41 | \( 1 + 30T + 1.68e3T^{2} \) |
| 43 | \( 1 + 2.82iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 67.8iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 60T + 2.80e3T^{2} \) |
| 59 | \( 1 + 42.4iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 28T + 3.72e3T^{2} \) |
| 67 | \( 1 - 82.0iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 56.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 10T + 5.32e3T^{2} \) |
| 79 | \( 1 + 113. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 25.4iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 22T + 7.92e3T^{2} \) |
| 97 | \( 1 - 150T + 9.40e3T^{2} \) |
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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
−9.402855138005648185733140638634, −8.515247026924144646871012321254, −7.67484450851540879447917889324, −6.86999329019866910000899011201, −5.94771889625528951630778354642, −5.28545940972996193274178889444, −4.72606280787076674608529276111, −3.28362910829426500296302983912, −2.22297005995156782541026144105, −1.83646567609845921241492403065,
0.26670275087806080278407233906, 1.23055585108132590394560203468, 2.55988177325439034870744817903, 3.43824380353790770137270608513, 4.48715743459351750163321803923, 5.20236194843611335452499509372, 6.18328611624371729491969128342, 6.90266002064276253083775873820, 7.61970694265595076208826305486, 8.354295589857998198789381741501
