| L(s) = 1 | − 4·4-s − 5i·7-s − 9·9-s + 3·11-s + 16·16-s + 15i·17-s + 19·19-s + 30i·23-s + 20i·28-s + 36·36-s + 85i·43-s − 12·44-s + 75i·47-s + 24·49-s + 103·61-s + ⋯ |
| L(s) = 1 | − 4-s − 0.714i·7-s − 9-s + 0.272·11-s + 16-s + 0.882i·17-s + 19-s + 1.30i·23-s + 0.714i·28-s + 36-s + 1.97i·43-s − 0.272·44-s + 1.59i·47-s + 0.489·49-s + 1.68·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.9715583521\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9715583521\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 19 | \( 1 - 19T \) |
| good | 2 | \( 1 + 4T^{2} \) |
| 3 | \( 1 + 9T^{2} \) |
| 7 | \( 1 + 5iT - 49T^{2} \) |
| 11 | \( 1 - 3T + 121T^{2} \) |
| 13 | \( 1 + 169T^{2} \) |
| 17 | \( 1 - 15iT - 289T^{2} \) |
| 23 | \( 1 - 30iT - 529T^{2} \) |
| 29 | \( 1 - 841T^{2} \) |
| 31 | \( 1 - 961T^{2} \) |
| 37 | \( 1 + 1.36e3T^{2} \) |
| 41 | \( 1 - 1.68e3T^{2} \) |
| 43 | \( 1 - 85iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 75iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 2.80e3T^{2} \) |
| 59 | \( 1 - 3.48e3T^{2} \) |
| 61 | \( 1 - 103T + 3.72e3T^{2} \) |
| 67 | \( 1 + 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 - 25iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 6.24e3T^{2} \) |
| 83 | \( 1 + 90iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 7.92e3T^{2} \) |
| 97 | \( 1 + 9.40e3T^{2} \) |
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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
−10.96348772132453140814658003016, −9.898551259455338163206544257748, −9.237904884078837185636814820836, −8.260771632350344332623621372625, −7.52347679468063298774791178230, −6.12877410451595404205008172589, −5.25321551886673374653250111097, −4.11342820987329293886349479990, −3.18221161227009083577586528517, −1.13332464440955089542180359985,
0.49320274841841272145180973859, 2.54783407496668986615495354907, 3.74022800459851719505078817882, 5.08250906432001103913579137753, 5.64315199029482536239989573967, 6.94912980040000317484386554876, 8.249656551299161358940077699160, 8.832656450362261840594529375692, 9.550768301795540069324048842231, 10.55417973163687070222589555231
