| L(s) = 1 | + 5·3-s − 5·5-s − 12·7-s − 2·9-s − 22·11-s + 19·13-s − 25·15-s + 96·17-s + 98·19-s − 60·21-s − 23·23-s + 25·25-s − 145·27-s − 227·29-s + 285·31-s − 110·33-s + 60·35-s − 398·37-s + 95·39-s + 271·41-s + 100·43-s + 10·45-s + 285·47-s − 199·49-s + 480·51-s + 18·53-s + 110·55-s + ⋯ |
| L(s) = 1 | + 0.962·3-s − 0.447·5-s − 0.647·7-s − 0.0740·9-s − 0.603·11-s + 0.405·13-s − 0.430·15-s + 1.36·17-s + 1.18·19-s − 0.623·21-s − 0.208·23-s + 1/5·25-s − 1.03·27-s − 1.45·29-s + 1.65·31-s − 0.580·33-s + 0.289·35-s − 1.76·37-s + 0.390·39-s + 1.03·41-s + 0.354·43-s + 0.0331·45-s + 0.884·47-s − 0.580·49-s + 1.31·51-s + 0.0466·53-s + 0.269·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + p T \) |
| 23 | \( 1 + p T \) |
| good | 3 | \( 1 - 5 T + p^{3} T^{2} \) |
| 7 | \( 1 + 12 T + p^{3} T^{2} \) |
| 11 | \( 1 + 2 p T + p^{3} T^{2} \) |
| 13 | \( 1 - 19 T + p^{3} T^{2} \) |
| 17 | \( 1 - 96 T + p^{3} T^{2} \) |
| 19 | \( 1 - 98 T + p^{3} T^{2} \) |
| 29 | \( 1 + 227 T + p^{3} T^{2} \) |
| 31 | \( 1 - 285 T + p^{3} T^{2} \) |
| 37 | \( 1 + 398 T + p^{3} T^{2} \) |
| 41 | \( 1 - 271 T + p^{3} T^{2} \) |
| 43 | \( 1 - 100 T + p^{3} T^{2} \) |
| 47 | \( 1 - 285 T + p^{3} T^{2} \) |
| 53 | \( 1 - 18 T + p^{3} T^{2} \) |
| 59 | \( 1 - 352 T + p^{3} T^{2} \) |
| 61 | \( 1 + 478 T + p^{3} T^{2} \) |
| 67 | \( 1 + 330 T + p^{3} T^{2} \) |
| 71 | \( 1 + 835 T + p^{3} T^{2} \) |
| 73 | \( 1 + 1127 T + p^{3} T^{2} \) |
| 79 | \( 1 + 322 T + p^{3} T^{2} \) |
| 83 | \( 1 + 572 T + p^{3} T^{2} \) |
| 89 | \( 1 + 504 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1712 T + p^{3} T^{2} \) |
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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
−8.483358440051538290155098492466, −7.70492238890796630857482571646, −7.28343357376945136303826436885, −5.98756872020852104863216489607, −5.35978095625069251181126707025, −4.05733444443759213162180763218, −3.24708808133730091016459790816, −2.76051227740709672765791072043, −1.34365631425879663511144373306, 0,
1.34365631425879663511144373306, 2.76051227740709672765791072043, 3.24708808133730091016459790816, 4.05733444443759213162180763218, 5.35978095625069251181126707025, 5.98756872020852104863216489607, 7.28343357376945136303826436885, 7.70492238890796630857482571646, 8.483358440051538290155098492466
