L(s) = 1 | − 5·5-s + 16·7-s − 24·11-s − 14·13-s + 18·17-s + 36·19-s − 104·23-s + 25·25-s + 250·29-s − 28·31-s − 80·35-s − 54·37-s − 354·41-s + 228·43-s − 408·47-s − 87·49-s − 262·53-s + 120·55-s + 64·59-s + 374·61-s + 70·65-s + 300·67-s − 1.01e3·71-s + 274·73-s − 384·77-s + 788·79-s + 396·83-s + ⋯ |
L(s) = 1 | − 0.447·5-s + 0.863·7-s − 0.657·11-s − 0.298·13-s + 0.256·17-s + 0.434·19-s − 0.942·23-s + 1/5·25-s + 1.60·29-s − 0.162·31-s − 0.386·35-s − 0.239·37-s − 1.34·41-s + 0.808·43-s − 1.26·47-s − 0.253·49-s − 0.679·53-s + 0.294·55-s + 0.141·59-s + 0.785·61-s + 0.133·65-s + 0.547·67-s − 1.69·71-s + 0.439·73-s − 0.568·77-s + 1.12·79-s + 0.523·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + p T \) |
good | 7 | \( 1 - 16 T + p^{3} T^{2} \) |
| 11 | \( 1 + 24 T + p^{3} T^{2} \) |
| 13 | \( 1 + 14 T + p^{3} T^{2} \) |
| 17 | \( 1 - 18 T + p^{3} T^{2} \) |
| 19 | \( 1 - 36 T + p^{3} T^{2} \) |
| 23 | \( 1 + 104 T + p^{3} T^{2} \) |
| 29 | \( 1 - 250 T + p^{3} T^{2} \) |
| 31 | \( 1 + 28 T + p^{3} T^{2} \) |
| 37 | \( 1 + 54 T + p^{3} T^{2} \) |
| 41 | \( 1 + 354 T + p^{3} T^{2} \) |
| 43 | \( 1 - 228 T + p^{3} T^{2} \) |
| 47 | \( 1 + 408 T + p^{3} T^{2} \) |
| 53 | \( 1 + 262 T + p^{3} T^{2} \) |
| 59 | \( 1 - 64 T + p^{3} T^{2} \) |
| 61 | \( 1 - 374 T + p^{3} T^{2} \) |
| 67 | \( 1 - 300 T + p^{3} T^{2} \) |
| 71 | \( 1 + 1016 T + p^{3} T^{2} \) |
| 73 | \( 1 - 274 T + p^{3} T^{2} \) |
| 79 | \( 1 - 788 T + p^{3} T^{2} \) |
| 83 | \( 1 - 396 T + p^{3} T^{2} \) |
| 89 | \( 1 + 786 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1086 T + p^{3} T^{2} \) |
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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
−8.467329379983277083320153500468, −8.069973264472887356272052356758, −7.29392201834200258595744675797, −6.32941905717651427389945483809, −5.22175418206935184668768722913, −4.66991757526515338218738024816, −3.56719507961430006996569248390, −2.50753651099500834346477758792, −1.33999268779207022759395489012, 0,
1.33999268779207022759395489012, 2.50753651099500834346477758792, 3.56719507961430006996569248390, 4.66991757526515338218738024816, 5.22175418206935184668768722913, 6.32941905717651427389945483809, 7.29392201834200258595744675797, 8.069973264472887356272052356758, 8.467329379983277083320153500468