| L(s) = 1 | − 2·2-s − 5·3-s + 4·4-s + 10·6-s − 8·8-s − 2·9-s − 57·11-s − 20·12-s − 70·13-s + 16·16-s + 51·17-s + 4·18-s − 5·19-s + 114·22-s − 69·23-s + 40·24-s + 140·26-s + 145·27-s + 114·29-s − 23·31-s − 32·32-s + 285·33-s − 102·34-s − 8·36-s + 253·37-s + 10·38-s + 350·39-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.962·3-s + 1/2·4-s + 0.680·6-s − 0.353·8-s − 0.0740·9-s − 1.56·11-s − 0.481·12-s − 1.49·13-s + 1/4·16-s + 0.727·17-s + 0.0523·18-s − 0.0603·19-s + 1.10·22-s − 0.625·23-s + 0.340·24-s + 1.05·26-s + 1.03·27-s + 0.729·29-s − 0.133·31-s − 0.176·32-s + 1.50·33-s − 0.514·34-s − 0.0370·36-s + 1.12·37-s + 0.0426·38-s + 1.43·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + p T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 5 T + p^{3} T^{2} \) |
| 11 | \( 1 + 57 T + p^{3} T^{2} \) |
| 13 | \( 1 + 70 T + p^{3} T^{2} \) |
| 17 | \( 1 - 3 p T + p^{3} T^{2} \) |
| 19 | \( 1 + 5 T + p^{3} T^{2} \) |
| 23 | \( 1 + 3 p T + p^{3} T^{2} \) |
| 29 | \( 1 - 114 T + p^{3} T^{2} \) |
| 31 | \( 1 + 23 T + p^{3} T^{2} \) |
| 37 | \( 1 - 253 T + p^{3} T^{2} \) |
| 41 | \( 1 - 42 T + p^{3} T^{2} \) |
| 43 | \( 1 - 124 T + p^{3} T^{2} \) |
| 47 | \( 1 - 201 T + p^{3} T^{2} \) |
| 53 | \( 1 - 393 T + p^{3} T^{2} \) |
| 59 | \( 1 + 219 T + p^{3} T^{2} \) |
| 61 | \( 1 - 709 T + p^{3} T^{2} \) |
| 67 | \( 1 + 419 T + p^{3} T^{2} \) |
| 71 | \( 1 + 96 T + p^{3} T^{2} \) |
| 73 | \( 1 + 313 T + p^{3} T^{2} \) |
| 79 | \( 1 - 461 T + p^{3} T^{2} \) |
| 83 | \( 1 + 588 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1017 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1834 T + p^{3} T^{2} \) |
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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
−8.021636236225360926920860299648, −7.59408244749594291282849694607, −6.74574680692852604687135982021, −5.75439265128907967302107963849, −5.32085380258185416639985022096, −4.43423920431735351018172630505, −2.91351105863432684193327522244, −2.30145299004886779966659437194, −0.78574602303638100180443059669, 0,
0.78574602303638100180443059669, 2.30145299004886779966659437194, 2.91351105863432684193327522244, 4.43423920431735351018172630505, 5.32085380258185416639985022096, 5.75439265128907967302107963849, 6.74574680692852604687135982021, 7.59408244749594291282849694607, 8.021636236225360926920860299648
