L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 8-s + 9-s − 5·11-s − 12-s − 13-s + 16-s + 2·17-s + 18-s + 7·19-s − 5·22-s − 3·23-s − 24-s − 26-s − 27-s − 6·31-s + 32-s + 5·33-s + 2·34-s + 36-s + 5·37-s + 7·38-s + 39-s − 9·41-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.353·8-s + 1/3·9-s − 1.50·11-s − 0.288·12-s − 0.277·13-s + 1/4·16-s + 0.485·17-s + 0.235·18-s + 1.60·19-s − 1.06·22-s − 0.625·23-s − 0.204·24-s − 0.196·26-s − 0.192·27-s − 1.07·31-s + 0.176·32-s + 0.870·33-s + 0.342·34-s + 1/6·36-s + 0.821·37-s + 1.13·38-s + 0.160·39-s − 1.40·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 13 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 - 10 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−7.55745906509988852700204459151, −6.83293087424457489750619369909, −5.90022823949632898161233424240, −5.33244640845193215965402588275, −5.01522010774876901743287002110, −3.99025819780955412740805026945, −3.18796240651000268410662963825, −2.42987325094402466578468394212, −1.34334206564400223147327397307, 0,
1.34334206564400223147327397307, 2.42987325094402466578468394212, 3.18796240651000268410662963825, 3.99025819780955412740805026945, 5.01522010774876901743287002110, 5.33244640845193215965402588275, 5.90022823949632898161233424240, 6.83293087424457489750619369909, 7.55745906509988852700204459151