| L(s) = 1 | + 4·3-s − 20·5-s − 11·9-s − 44·11-s − 44·13-s − 80·15-s + 72·17-s − 100·19-s + 120·23-s + 275·25-s − 152·27-s + 218·29-s + 280·31-s − 176·33-s − 30·37-s − 176·39-s + 120·41-s − 220·43-s + 220·45-s − 88·47-s + 288·51-s + 110·53-s + 880·55-s − 400·57-s − 580·59-s + 380·61-s + 880·65-s + ⋯ |
| L(s) = 1 | + 0.769·3-s − 1.78·5-s − 0.407·9-s − 1.20·11-s − 0.938·13-s − 1.37·15-s + 1.02·17-s − 1.20·19-s + 1.08·23-s + 11/5·25-s − 1.08·27-s + 1.39·29-s + 1.62·31-s − 0.928·33-s − 0.133·37-s − 0.722·39-s + 0.457·41-s − 0.780·43-s + 0.728·45-s − 0.273·47-s + 0.790·51-s + 0.285·53-s + 2.15·55-s − 0.929·57-s − 1.27·59-s + 0.797·61-s + 1.67·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.120098524\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.120098524\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 4 T + p^{3} T^{2} \) |
| 5 | \( 1 + 4 p T + p^{3} T^{2} \) |
| 11 | \( 1 + 4 p T + p^{3} T^{2} \) |
| 13 | \( 1 + 44 T + p^{3} T^{2} \) |
| 17 | \( 1 - 72 T + p^{3} T^{2} \) |
| 19 | \( 1 + 100 T + p^{3} T^{2} \) |
| 23 | \( 1 - 120 T + p^{3} T^{2} \) |
| 29 | \( 1 - 218 T + p^{3} T^{2} \) |
| 31 | \( 1 - 280 T + p^{3} T^{2} \) |
| 37 | \( 1 + 30 T + p^{3} T^{2} \) |
| 41 | \( 1 - 120 T + p^{3} T^{2} \) |
| 43 | \( 1 + 220 T + p^{3} T^{2} \) |
| 47 | \( 1 + 88 T + p^{3} T^{2} \) |
| 53 | \( 1 - 110 T + p^{3} T^{2} \) |
| 59 | \( 1 + 580 T + p^{3} T^{2} \) |
| 61 | \( 1 - 380 T + p^{3} T^{2} \) |
| 67 | \( 1 - 980 T + p^{3} T^{2} \) |
| 71 | \( 1 - 112 T + p^{3} T^{2} \) |
| 73 | \( 1 + 640 T + p^{3} T^{2} \) |
| 79 | \( 1 - 488 T + p^{3} T^{2} \) |
| 83 | \( 1 + 660 T + p^{3} T^{2} \) |
| 89 | \( 1 - 320 T + p^{3} T^{2} \) |
| 97 | \( 1 - 248 T + p^{3} T^{2} \) |
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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
−9.947517830905665058055377690328, −8.678064585616852138022003734496, −8.142559882280752243724061073978, −7.66651653951383544577476051083, −6.69169789266608768326034642217, −5.16168703919447053124665574664, −4.37864206975188475772503012217, −3.20167121825912417755793001120, −2.63444769422802877644880971013, −0.54546982228488694110514402646,
0.54546982228488694110514402646, 2.63444769422802877644880971013, 3.20167121825912417755793001120, 4.37864206975188475772503012217, 5.16168703919447053124665574664, 6.69169789266608768326034642217, 7.66651653951383544577476051083, 8.142559882280752243724061073978, 8.678064585616852138022003734496, 9.947517830905665058055377690328
