| L(s) = 1 | − 1.41·3-s − 5-s − 2.82·7-s − 0.999·9-s − 4.82·11-s + 0.585·13-s + 1.41·15-s − 0.828·17-s + 19-s + 4.00·21-s − 4·23-s + 25-s + 5.65·27-s + 4.82·29-s + 6.82·33-s + 2.82·35-s + 1.75·37-s − 0.828·39-s + 4.82·41-s + 2.82·43-s + 0.999·45-s − 8.48·47-s + 1.00·49-s + 1.17·51-s − 1.07·53-s + 4.82·55-s − 1.41·57-s + ⋯ |
| L(s) = 1 | − 0.816·3-s − 0.447·5-s − 1.06·7-s − 0.333·9-s − 1.45·11-s + 0.162·13-s + 0.365·15-s − 0.200·17-s + 0.229·19-s + 0.872·21-s − 0.834·23-s + 0.200·25-s + 1.08·27-s + 0.896·29-s + 1.18·33-s + 0.478·35-s + 0.288·37-s − 0.132·39-s + 0.754·41-s + 0.431·43-s + 0.149·45-s − 1.23·47-s + 0.142·49-s + 0.164·51-s − 0.147·53-s + 0.651·55-s − 0.187·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5234563313\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5234563313\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 1.41T + 3T^{2} \) |
| 7 | \( 1 + 2.82T + 7T^{2} \) |
| 11 | \( 1 + 4.82T + 11T^{2} \) |
| 13 | \( 1 - 0.585T + 13T^{2} \) |
| 17 | \( 1 + 0.828T + 17T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 - 4.82T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 1.75T + 37T^{2} \) |
| 41 | \( 1 - 4.82T + 41T^{2} \) |
| 43 | \( 1 - 2.82T + 43T^{2} \) |
| 47 | \( 1 + 8.48T + 47T^{2} \) |
| 53 | \( 1 + 1.07T + 53T^{2} \) |
| 59 | \( 1 - 2.82T + 59T^{2} \) |
| 61 | \( 1 - 9.65T + 61T^{2} \) |
| 67 | \( 1 - 6.58T + 67T^{2} \) |
| 71 | \( 1 - 7.31T + 71T^{2} \) |
| 73 | \( 1 + 16.1T + 73T^{2} \) |
| 79 | \( 1 - 14.8T + 79T^{2} \) |
| 83 | \( 1 + 8T + 83T^{2} \) |
| 89 | \( 1 + 8.82T + 89T^{2} \) |
| 97 | \( 1 - 6.24T + 97T^{2} \) |
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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
−9.678396174006840221426253358788, −8.546023845162878341187752307343, −7.901800735337600638582187594710, −6.90713023617083646657899514980, −6.14750277908761481683413296837, −5.44996691928555218222450854756, −4.55945215810712469391339379799, −3.36684813268620376838480250098, −2.51801468018039126786721732517, −0.50151682254725627847781573475,
0.50151682254725627847781573475, 2.51801468018039126786721732517, 3.36684813268620376838480250098, 4.55945215810712469391339379799, 5.44996691928555218222450854756, 6.14750277908761481683413296837, 6.90713023617083646657899514980, 7.901800735337600638582187594710, 8.546023845162878341187752307343, 9.678396174006840221426253358788
