| L(s) = 1 | − 0.302·3-s + 5-s + 3.30·7-s − 2.90·9-s + 0.302·11-s − 1.30·13-s − 0.302·15-s + 3.69·17-s − 7.90·19-s − 1.00·21-s − 23-s + 25-s + 1.78·27-s − 1.39·29-s + 0.697·31-s − 0.0916·33-s + 3.30·35-s + 9.21·37-s + 0.394·39-s − 8.30·41-s + 4·43-s − 2.90·45-s + 10.6·47-s + 3.90·49-s − 1.11·51-s − 7.21·53-s + 0.302·55-s + ⋯ |
| L(s) = 1 | − 0.174·3-s + 0.447·5-s + 1.24·7-s − 0.969·9-s + 0.0912·11-s − 0.361·13-s − 0.0781·15-s + 0.896·17-s − 1.81·19-s − 0.218·21-s − 0.208·23-s + 0.200·25-s + 0.344·27-s − 0.258·29-s + 0.125·31-s − 0.0159·33-s + 0.558·35-s + 1.51·37-s + 0.0631·39-s − 1.29·41-s + 0.609·43-s − 0.433·45-s + 1.54·47-s + 0.558·49-s − 0.156·51-s − 0.990·53-s + 0.0408·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.092996764\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.092996764\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 + 0.302T + 3T^{2} \) |
| 7 | \( 1 - 3.30T + 7T^{2} \) |
| 11 | \( 1 - 0.302T + 11T^{2} \) |
| 13 | \( 1 + 1.30T + 13T^{2} \) |
| 17 | \( 1 - 3.69T + 17T^{2} \) |
| 19 | \( 1 + 7.90T + 19T^{2} \) |
| 29 | \( 1 + 1.39T + 29T^{2} \) |
| 31 | \( 1 - 0.697T + 31T^{2} \) |
| 37 | \( 1 - 9.21T + 37T^{2} \) |
| 41 | \( 1 + 8.30T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 - 10.6T + 47T^{2} \) |
| 53 | \( 1 + 7.21T + 53T^{2} \) |
| 59 | \( 1 - 8.60T + 59T^{2} \) |
| 61 | \( 1 - 2.30T + 61T^{2} \) |
| 67 | \( 1 + 5.21T + 67T^{2} \) |
| 71 | \( 1 - 1.69T + 71T^{2} \) |
| 73 | \( 1 - 12.6T + 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 + 3.21T + 83T^{2} \) |
| 89 | \( 1 - 17.2T + 89T^{2} \) |
| 97 | \( 1 - 18.7T + 97T^{2} \) |
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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
−8.002912225095064904277172679203, −7.31305912939638893808623971475, −6.29333969318942529811688635287, −5.86919248505256673419476323201, −5.06345729238177616434439874867, −4.52333517359616098617612015751, −3.57643016570454999220952286383, −2.47927197662107214792109232126, −1.92251407453233566532867499728, −0.73150734587939924496508066816,
0.73150734587939924496508066816, 1.92251407453233566532867499728, 2.47927197662107214792109232126, 3.57643016570454999220952286383, 4.52333517359616098617612015751, 5.06345729238177616434439874867, 5.86919248505256673419476323201, 6.29333969318942529811688635287, 7.31305912939638893808623971475, 8.002912225095064904277172679203
