| L(s) = 1 | + 4·3-s − 16·7-s − 11·9-s + 60·11-s − 86·13-s − 18·17-s − 44·19-s − 64·21-s + 48·23-s − 152·27-s − 186·29-s − 176·31-s + 240·33-s − 254·37-s − 344·39-s + 186·41-s − 100·43-s + 168·47-s − 87·49-s − 72·51-s + 498·53-s − 176·57-s + 252·59-s − 58·61-s + 176·63-s − 1.03e3·67-s + 192·69-s + ⋯ |
| L(s) = 1 | + 0.769·3-s − 0.863·7-s − 0.407·9-s + 1.64·11-s − 1.83·13-s − 0.256·17-s − 0.531·19-s − 0.665·21-s + 0.435·23-s − 1.08·27-s − 1.19·29-s − 1.01·31-s + 1.26·33-s − 1.12·37-s − 1.41·39-s + 0.708·41-s − 0.354·43-s + 0.521·47-s − 0.253·49-s − 0.197·51-s + 1.29·53-s − 0.408·57-s + 0.556·59-s − 0.121·61-s + 0.351·63-s − 1.88·67-s + 0.334·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 3 | \( 1 - 4 T + p^{3} T^{2} \) |
| 7 | \( 1 + 16 T + p^{3} T^{2} \) |
| 11 | \( 1 - 60 T + p^{3} T^{2} \) |
| 13 | \( 1 + 86 T + p^{3} T^{2} \) |
| 17 | \( 1 + 18 T + p^{3} T^{2} \) |
| 19 | \( 1 + 44 T + p^{3} T^{2} \) |
| 23 | \( 1 - 48 T + p^{3} T^{2} \) |
| 29 | \( 1 + 186 T + p^{3} T^{2} \) |
| 31 | \( 1 + 176 T + p^{3} T^{2} \) |
| 37 | \( 1 + 254 T + p^{3} T^{2} \) |
| 41 | \( 1 - 186 T + p^{3} T^{2} \) |
| 43 | \( 1 + 100 T + p^{3} T^{2} \) |
| 47 | \( 1 - 168 T + p^{3} T^{2} \) |
| 53 | \( 1 - 498 T + p^{3} T^{2} \) |
| 59 | \( 1 - 252 T + p^{3} T^{2} \) |
| 61 | \( 1 + 58 T + p^{3} T^{2} \) |
| 67 | \( 1 + 1036 T + p^{3} T^{2} \) |
| 71 | \( 1 + 168 T + p^{3} T^{2} \) |
| 73 | \( 1 + 506 T + p^{3} T^{2} \) |
| 79 | \( 1 + 272 T + p^{3} T^{2} \) |
| 83 | \( 1 - 948 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1014 T + p^{3} T^{2} \) |
| 97 | \( 1 - 766 T + p^{3} T^{2} \) |
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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
−10.16426665220507567247807451153, −9.215984551634522830672230320001, −8.933546350909068376610617337754, −7.49369100945417328689225532375, −6.78214162283973045706533823123, −5.60785756799232395335959444254, −4.17667124406819898322039571891, −3.17944034026101807251626063054, −2.02399032199331437437601211593, 0,
2.02399032199331437437601211593, 3.17944034026101807251626063054, 4.17667124406819898322039571891, 5.60785756799232395335959444254, 6.78214162283973045706533823123, 7.49369100945417328689225532375, 8.933546350909068376610617337754, 9.215984551634522830672230320001, 10.16426665220507567247807451153
