L(s) = 1 | − 3-s − 2·7-s + 9-s − 4·11-s − 13-s + 17-s + 2·21-s + 8·23-s − 27-s + 6·29-s + 2·31-s + 4·33-s − 4·37-s + 39-s − 2·41-s − 4·43-s + 2·47-s − 3·49-s − 51-s − 10·53-s − 10·59-s + 14·61-s − 2·63-s − 12·67-s − 8·69-s + 8·71-s − 8·73-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.755·7-s + 1/3·9-s − 1.20·11-s − 0.277·13-s + 0.242·17-s + 0.436·21-s + 1.66·23-s − 0.192·27-s + 1.11·29-s + 0.359·31-s + 0.696·33-s − 0.657·37-s + 0.160·39-s − 0.312·41-s − 0.609·43-s + 0.291·47-s − 3/7·49-s − 0.140·51-s − 1.37·53-s − 1.30·59-s + 1.79·61-s − 0.251·63-s − 1.46·67-s − 0.963·69-s + 0.949·71-s − 0.936·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 265200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 265200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8963489742\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8963489742\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 - T \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−12.83943125602693, −12.38177674350631, −11.90274011700430, −11.46720750689241, −10.78365014950355, −10.57159956443638, −10.13350944435999, −9.607605136294510, −9.209234278188286, −8.593906270201475, −8.072047258028582, −7.641140324777726, −7.016528205176217, −6.658781658002780, −6.252799610835457, −5.536395729113923, −5.158575912135742, −4.763906024741018, −4.219992283273907, −3.253282051624654, −3.114817515398658, −2.509927578436376, −1.713448200978329, −0.9982202364369472, −0.3037546373351609,
0.3037546373351609, 0.9982202364369472, 1.713448200978329, 2.509927578436376, 3.114817515398658, 3.253282051624654, 4.219992283273907, 4.763906024741018, 5.158575912135742, 5.536395729113923, 6.252799610835457, 6.658781658002780, 7.016528205176217, 7.641140324777726, 8.072047258028582, 8.593906270201475, 9.209234278188286, 9.607605136294510, 10.13350944435999, 10.57159956443638, 10.78365014950355, 11.46720750689241, 11.90274011700430, 12.38177674350631, 12.83943125602693