| L(s) = 1 | − 2.90·3-s + 12.8·5-s + 11.9·7-s − 18.5·9-s − 48.1·11-s − 8.87·13-s − 37.2·15-s − 105.·17-s + 19·19-s − 34.8·21-s + 112.·23-s + 39.4·25-s + 132.·27-s − 0.592·29-s + 71.6·31-s + 139.·33-s + 153.·35-s + 199.·37-s + 25.7·39-s + 16.4·41-s + 516.·43-s − 238.·45-s − 68.2·47-s − 199.·49-s + 307.·51-s + 683.·53-s − 617.·55-s + ⋯ |
| L(s) = 1 | − 0.558·3-s + 1.14·5-s + 0.647·7-s − 0.687·9-s − 1.32·11-s − 0.189·13-s − 0.641·15-s − 1.51·17-s + 0.229·19-s − 0.362·21-s + 1.01·23-s + 0.315·25-s + 0.943·27-s − 0.00379·29-s + 0.414·31-s + 0.737·33-s + 0.742·35-s + 0.885·37-s + 0.105·39-s + 0.0626·41-s + 1.83·43-s − 0.788·45-s − 0.211·47-s − 0.580·49-s + 0.844·51-s + 1.77·53-s − 1.51·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.751113320\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.751113320\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 - 19T \) |
| good | 3 | \( 1 + 2.90T + 27T^{2} \) |
| 5 | \( 1 - 12.8T + 125T^{2} \) |
| 7 | \( 1 - 11.9T + 343T^{2} \) |
| 11 | \( 1 + 48.1T + 1.33e3T^{2} \) |
| 13 | \( 1 + 8.87T + 2.19e3T^{2} \) |
| 17 | \( 1 + 105.T + 4.91e3T^{2} \) |
| 23 | \( 1 - 112.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 0.592T + 2.43e4T^{2} \) |
| 31 | \( 1 - 71.6T + 2.97e4T^{2} \) |
| 37 | \( 1 - 199.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 16.4T + 6.89e4T^{2} \) |
| 43 | \( 1 - 516.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 68.2T + 1.03e5T^{2} \) |
| 53 | \( 1 - 683.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 140.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 433.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 375.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 864.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 379.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 879.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.25e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 91.4T + 7.04e5T^{2} \) |
| 97 | \( 1 + 96.8T + 9.12e5T^{2} \) |
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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
−9.350258533937178812211032926104, −8.631829942284266316750505923286, −7.73396371844387816855758489197, −6.71437806094084297409900836664, −5.84270313719254308021217613980, −5.25263311031345096950641464532, −4.51057261324440383313521377732, −2.77434104094170023045184183667, −2.14359579074855441153440107654, −0.67634476647857537229120684312,
0.67634476647857537229120684312, 2.14359579074855441153440107654, 2.77434104094170023045184183667, 4.51057261324440383313521377732, 5.25263311031345096950641464532, 5.84270313719254308021217613980, 6.71437806094084297409900836664, 7.73396371844387816855758489197, 8.631829942284266316750505923286, 9.350258533937178812211032926104
