| L(s) = 1 | + 4·3-s − 5·5-s + 16·7-s − 11·9-s − 60·11-s − 86·13-s − 20·15-s + 18·17-s + 44·19-s + 64·21-s − 48·23-s + 25·25-s − 152·27-s + 186·29-s − 176·31-s − 240·33-s − 80·35-s − 254·37-s − 344·39-s + 186·41-s − 100·43-s + 55·45-s − 168·47-s − 87·49-s + 72·51-s + 498·53-s + 300·55-s + ⋯ |
| L(s) = 1 | + 0.769·3-s − 0.447·5-s + 0.863·7-s − 0.407·9-s − 1.64·11-s − 1.83·13-s − 0.344·15-s + 0.256·17-s + 0.531·19-s + 0.665·21-s − 0.435·23-s + 1/5·25-s − 1.08·27-s + 1.19·29-s − 1.01·31-s − 1.26·33-s − 0.386·35-s − 1.12·37-s − 1.41·39-s + 0.708·41-s − 0.354·43-s + 0.182·45-s − 0.521·47-s − 0.253·49-s + 0.197·51-s + 1.29·53-s + 0.735·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + p T \) |
| 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
−10.66594512393171096976715746833, −9.819685119366186953620381150653, −8.641981090746485913195025934616, −7.83387058979112267978832495539, −7.35262018419568046800162158390, −5.46104685886736572930851625156, −4.71857606343080787150891611703, −3.11671469686772335442604811164, −2.20019938955047118294615154473, 0,
2.20019938955047118294615154473, 3.11671469686772335442604811164, 4.71857606343080787150891611703, 5.46104685886736572930851625156, 7.35262018419568046800162158390, 7.83387058979112267978832495539, 8.641981090746485913195025934616, 9.819685119366186953620381150653, 10.66594512393171096976715746833
