| L(s) = 1 | − 3-s − 5-s + 4·7-s + 9-s + 13-s + 15-s + 3·19-s − 4·21-s − 2·23-s + 25-s − 27-s − 3·29-s − 4·31-s − 4·35-s + 10·37-s − 39-s + 5·41-s − 8·43-s − 45-s + 7·47-s + 9·49-s − 3·53-s − 3·57-s − 8·61-s + 4·63-s − 65-s − 5·67-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1.51·7-s + 1/3·9-s + 0.277·13-s + 0.258·15-s + 0.688·19-s − 0.872·21-s − 0.417·23-s + 1/5·25-s − 0.192·27-s − 0.557·29-s − 0.718·31-s − 0.676·35-s + 1.64·37-s − 0.160·39-s + 0.780·41-s − 1.21·43-s − 0.149·45-s + 1.02·47-s + 9/7·49-s − 0.412·53-s − 0.397·57-s − 1.02·61-s + 0.503·63-s − 0.124·65-s − 0.610·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 188760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 188760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.360050101\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.360050101\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 - 13 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−12.83290666320260, −12.77363608626255, −12.00353752467254, −11.55055602554813, −11.34758792344305, −10.95484947763494, −10.40832780648545, −9.918822286637164, −9.188264089342581, −8.947181351884063, −8.068533851099086, −7.919671534420034, −7.458969613205011, −6.936051596004495, −6.206607440061392, −5.781660171857091, −5.214020108565841, −4.815078194183369, −4.243586107535136, −3.841253198670173, −3.111979092521908, −2.341983278480102, −1.717534556460774, −1.163799622570592, −0.4962957331858044,
0.4962957331858044, 1.163799622570592, 1.717534556460774, 2.341983278480102, 3.111979092521908, 3.841253198670173, 4.243586107535136, 4.815078194183369, 5.214020108565841, 5.781660171857091, 6.206607440061392, 6.936051596004495, 7.458969613205011, 7.919671534420034, 8.068533851099086, 8.947181351884063, 9.188264089342581, 9.918822286637164, 10.40832780648545, 10.95484947763494, 11.34758792344305, 11.55055602554813, 12.00353752467254, 12.77363608626255, 12.83290666320260
