L(s) = 1 | + 15.5·3-s − 96.7·5-s − 0.445·9-s − 281.·11-s + 269.·13-s − 1.50e3·15-s − 1.71e3·17-s − 1.17e3·19-s − 785.·23-s + 6.23e3·25-s − 3.79e3·27-s + 6.14e3·29-s − 7.00e3·31-s − 4.38e3·33-s − 1.14e4·37-s + 4.19e3·39-s + 1.32e4·41-s + 1.98e4·43-s + 43.0·45-s + 6.88e3·47-s − 2.67e4·51-s + 8.65e3·53-s + 2.72e4·55-s − 1.82e4·57-s − 4.78e4·59-s − 5.27e4·61-s − 2.60e4·65-s + ⋯ |
L(s) = 1 | + 0.999·3-s − 1.73·5-s − 0.00183·9-s − 0.701·11-s + 0.442·13-s − 1.72·15-s − 1.44·17-s − 0.745·19-s − 0.309·23-s + 1.99·25-s − 1.00·27-s + 1.35·29-s − 1.30·31-s − 0.700·33-s − 1.38·37-s + 0.441·39-s + 1.23·41-s + 1.63·43-s + 0.00316·45-s + 0.454·47-s − 1.44·51-s + 0.423·53-s + 1.21·55-s − 0.744·57-s − 1.78·59-s − 1.81·61-s − 0.764·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(1.067289831\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.067289831\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 - 15.5T + 243T^{2} \) |
| 5 | \( 1 + 96.7T + 3.12e3T^{2} \) |
| 11 | \( 1 + 281.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 269.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.71e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.17e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 785.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 6.14e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 7.00e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.14e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.32e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.98e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 6.88e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 8.65e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 4.78e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 5.27e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 2.40e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 1.25e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 4.07e3T + 2.07e9T^{2} \) |
| 79 | \( 1 - 1.19e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 8.19e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 9.69e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 2.64e4T + 8.58e9T^{2} \) |
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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
−9.048476159367070285011801368275, −8.708988968091670801574101605657, −7.86269253868317674735246884248, −7.34387618603280811510948463440, −6.17365756438893341067619383461, −4.69699497838588619280626293231, −3.96858360510553881022200030206, −3.12245458882846608052616051141, −2.16352559764325482626637631867, −0.42531358218440013683530321558,
0.42531358218440013683530321558, 2.16352559764325482626637631867, 3.12245458882846608052616051141, 3.96858360510553881022200030206, 4.69699497838588619280626293231, 6.17365756438893341067619383461, 7.34387618603280811510948463440, 7.86269253868317674735246884248, 8.708988968091670801574101605657, 9.048476159367070285011801368275