| L(s) = 1 | + 2.67·2-s − 3-s + 5.14·4-s − 1.32·5-s − 2.67·6-s − 7-s + 8.39·8-s + 9-s − 3.54·10-s − 3.03·11-s − 5.14·12-s − 2.67·14-s + 1.32·15-s + 12.1·16-s + 4.56·17-s + 2.67·18-s + 0.940·19-s − 6.81·20-s + 21-s − 8.10·22-s + 8.55·23-s − 8.39·24-s − 3.24·25-s − 27-s − 5.14·28-s + 5.79·29-s + 3.54·30-s + ⋯ |
| L(s) = 1 | + 1.88·2-s − 0.577·3-s + 2.57·4-s − 0.592·5-s − 1.09·6-s − 0.377·7-s + 2.96·8-s + 0.333·9-s − 1.12·10-s − 0.914·11-s − 1.48·12-s − 0.714·14-s + 0.342·15-s + 3.03·16-s + 1.10·17-s + 0.629·18-s + 0.215·19-s − 1.52·20-s + 0.218·21-s − 1.72·22-s + 1.78·23-s − 1.71·24-s − 0.648·25-s − 0.192·27-s − 0.971·28-s + 1.07·29-s + 0.646·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.875657291\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.875657291\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 2.67T + 2T^{2} \) |
| 5 | \( 1 + 1.32T + 5T^{2} \) |
| 11 | \( 1 + 3.03T + 11T^{2} \) |
| 17 | \( 1 - 4.56T + 17T^{2} \) |
| 19 | \( 1 - 0.940T + 19T^{2} \) |
| 23 | \( 1 - 8.55T + 23T^{2} \) |
| 29 | \( 1 - 5.79T + 29T^{2} \) |
| 31 | \( 1 - 1.00T + 31T^{2} \) |
| 37 | \( 1 - 11.3T + 37T^{2} \) |
| 41 | \( 1 + 7.71T + 41T^{2} \) |
| 43 | \( 1 - 7.20T + 43T^{2} \) |
| 47 | \( 1 - 12.8T + 47T^{2} \) |
| 53 | \( 1 - 5.26T + 53T^{2} \) |
| 59 | \( 1 + 13.1T + 59T^{2} \) |
| 61 | \( 1 - 5.82T + 61T^{2} \) |
| 67 | \( 1 - 3.02T + 67T^{2} \) |
| 71 | \( 1 + 6.04T + 71T^{2} \) |
| 73 | \( 1 + 12.2T + 73T^{2} \) |
| 79 | \( 1 - 1.46T + 79T^{2} \) |
| 83 | \( 1 - 4.31T + 83T^{2} \) |
| 89 | \( 1 + 3.22T + 89T^{2} \) |
| 97 | \( 1 + 5.51T + 97T^{2} \) |
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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
−8.145744633347228274537530819924, −7.42386572094822953127338468038, −6.93385640264947015580935375116, −5.95461805272815655472022756376, −5.52477685716629592203364250299, −4.73424368153139595132488182418, −4.10198665114468585642593501732, −3.14473230843182922620395825715, −2.60630318261803059938219132984, −1.04717610594165875993501620301,
1.04717610594165875993501620301, 2.60630318261803059938219132984, 3.14473230843182922620395825715, 4.10198665114468585642593501732, 4.73424368153139595132488182418, 5.52477685716629592203364250299, 5.95461805272815655472022756376, 6.93385640264947015580935375116, 7.42386572094822953127338468038, 8.145744633347228274537530819924
