| L(s) = 1 | − 2·2-s + 4·4-s − 14·7-s − 8·8-s − 6·11-s − 68·13-s + 28·14-s + 16·16-s + 78·17-s + 44·19-s + 12·22-s + 120·23-s + 136·26-s − 56·28-s − 126·29-s − 244·31-s − 32·32-s − 156·34-s + 304·37-s − 88·38-s + 480·41-s − 104·43-s − 24·44-s − 240·46-s + 600·47-s − 147·49-s − 272·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.755·7-s − 0.353·8-s − 0.164·11-s − 1.45·13-s + 0.534·14-s + 1/4·16-s + 1.11·17-s + 0.531·19-s + 0.116·22-s + 1.08·23-s + 1.02·26-s − 0.377·28-s − 0.806·29-s − 1.41·31-s − 0.176·32-s − 0.786·34-s + 1.35·37-s − 0.375·38-s + 1.82·41-s − 0.368·43-s − 0.0822·44-s − 0.769·46-s + 1.86·47-s − 3/7·49-s − 0.725·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.043471466\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.043471466\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 2 p T + p^{3} T^{2} \) |
| 11 | \( 1 + 6 T + p^{3} T^{2} \) |
| 13 | \( 1 + 68 T + p^{3} T^{2} \) |
| 17 | \( 1 - 78 T + p^{3} T^{2} \) |
| 19 | \( 1 - 44 T + p^{3} T^{2} \) |
| 23 | \( 1 - 120 T + p^{3} T^{2} \) |
| 29 | \( 1 + 126 T + p^{3} T^{2} \) |
| 31 | \( 1 + 244 T + p^{3} T^{2} \) |
| 37 | \( 1 - 304 T + p^{3} T^{2} \) |
| 41 | \( 1 - 480 T + p^{3} T^{2} \) |
| 43 | \( 1 + 104 T + p^{3} T^{2} \) |
| 47 | \( 1 - 600 T + p^{3} T^{2} \) |
| 53 | \( 1 + 258 T + p^{3} T^{2} \) |
| 59 | \( 1 + 534 T + p^{3} T^{2} \) |
| 61 | \( 1 - 362 T + p^{3} T^{2} \) |
| 67 | \( 1 - 4 p T + p^{3} T^{2} \) |
| 71 | \( 1 - 972 T + p^{3} T^{2} \) |
| 73 | \( 1 + 470 T + p^{3} T^{2} \) |
| 79 | \( 1 - 1244 T + p^{3} T^{2} \) |
| 83 | \( 1 - 396 T + p^{3} T^{2} \) |
| 89 | \( 1 - 972 T + p^{3} T^{2} \) |
| 97 | \( 1 - 46 T + p^{3} T^{2} \) |
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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
−10.52873448966348334593758137999, −9.527384514854369337038634369028, −9.263081368238322834588518547265, −7.69430038399690922992765784190, −7.33122259617803236105047640323, −6.07154081068786469847128675451, −5.06500632155895097057536809007, −3.47009662258862233352870022221, −2.39179668370222691032868037628, −0.70456975136585232026115801955,
0.70456975136585232026115801955, 2.39179668370222691032868037628, 3.47009662258862233352870022221, 5.06500632155895097057536809007, 6.07154081068786469847128675451, 7.33122259617803236105047640323, 7.69430038399690922992765784190, 9.263081368238322834588518547265, 9.527384514854369337038634369028, 10.52873448966348334593758137999
