| L(s) = 1 | − 4·3-s − 24·7-s − 11·9-s + 44·11-s + 22·13-s − 50·17-s − 44·19-s + 96·21-s + 56·23-s + 152·27-s − 198·29-s − 160·31-s − 176·33-s − 162·37-s − 88·39-s − 198·41-s + 52·43-s − 528·47-s + 233·49-s + 200·51-s − 242·53-s + 176·57-s + 668·59-s − 550·61-s + 264·63-s + 188·67-s − 224·69-s + ⋯ |
| L(s) = 1 | − 0.769·3-s − 1.29·7-s − 0.407·9-s + 1.20·11-s + 0.469·13-s − 0.713·17-s − 0.531·19-s + 0.997·21-s + 0.507·23-s + 1.08·27-s − 1.26·29-s − 0.926·31-s − 0.928·33-s − 0.719·37-s − 0.361·39-s − 0.754·41-s + 0.184·43-s − 1.63·47-s + 0.679·49-s + 0.549·51-s − 0.627·53-s + 0.408·57-s + 1.47·59-s − 1.15·61-s + 0.527·63-s + 0.342·67-s − 0.390·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.6982624422\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6982624422\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 4 T + p^{3} T^{2} \) |
| 7 | \( 1 + 24 T + p^{3} T^{2} \) |
| 11 | \( 1 - 4 p T + p^{3} T^{2} \) |
| 13 | \( 1 - 22 T + p^{3} T^{2} \) |
| 17 | \( 1 + 50 T + p^{3} T^{2} \) |
| 19 | \( 1 + 44 T + p^{3} T^{2} \) |
| 23 | \( 1 - 56 T + p^{3} T^{2} \) |
| 29 | \( 1 + 198 T + p^{3} T^{2} \) |
| 31 | \( 1 + 160 T + p^{3} T^{2} \) |
| 37 | \( 1 + 162 T + p^{3} T^{2} \) |
| 41 | \( 1 + 198 T + p^{3} T^{2} \) |
| 43 | \( 1 - 52 T + p^{3} T^{2} \) |
| 47 | \( 1 + 528 T + p^{3} T^{2} \) |
| 53 | \( 1 + 242 T + p^{3} T^{2} \) |
| 59 | \( 1 - 668 T + p^{3} T^{2} \) |
| 61 | \( 1 + 550 T + p^{3} T^{2} \) |
| 67 | \( 1 - 188 T + p^{3} T^{2} \) |
| 71 | \( 1 - 728 T + p^{3} T^{2} \) |
| 73 | \( 1 + 154 T + p^{3} T^{2} \) |
| 79 | \( 1 + 656 T + p^{3} T^{2} \) |
| 83 | \( 1 - 236 T + p^{3} T^{2} \) |
| 89 | \( 1 - 714 T + p^{3} T^{2} \) |
| 97 | \( 1 - 478 T + p^{3} T^{2} \) |
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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
−9.107709323860641581900832293203, −8.464538737483824747568340812583, −7.08533394383244434329191690844, −6.51655928678710411893785647537, −5.98928162117284932912307185249, −5.03658305607008634299537931237, −3.87803097277403402533073699968, −3.19559054938594829146639420570, −1.78850031024197776278634762393, −0.40983023410998793016771975073,
0.40983023410998793016771975073, 1.78850031024197776278634762393, 3.19559054938594829146639420570, 3.87803097277403402533073699968, 5.03658305607008634299537931237, 5.98928162117284932912307185249, 6.51655928678710411893785647537, 7.08533394383244434329191690844, 8.464538737483824747568340812583, 9.107709323860641581900832293203
