| L(s) = 1 | − 5.83·3-s + 12.0·5-s + 7·7-s + 7.02·9-s − 11·11-s + 31.7·13-s − 70.1·15-s + 112.·17-s + 6.19·19-s − 40.8·21-s − 113.·23-s + 19.5·25-s + 116.·27-s + 71.8·29-s + 88.9·31-s + 64.1·33-s + 84.1·35-s − 92.9·37-s − 184.·39-s + 192.·41-s − 124.·43-s + 84.4·45-s + 272.·47-s + 49·49-s − 653.·51-s − 122.·53-s − 132.·55-s + ⋯ |
| L(s) = 1 | − 1.12·3-s + 1.07·5-s + 0.377·7-s + 0.260·9-s − 0.301·11-s + 0.676·13-s − 1.20·15-s + 1.59·17-s + 0.0747·19-s − 0.424·21-s − 1.03·23-s + 0.156·25-s + 0.830·27-s + 0.459·29-s + 0.515·31-s + 0.338·33-s + 0.406·35-s − 0.412·37-s − 0.759·39-s + 0.733·41-s − 0.441·43-s + 0.279·45-s + 0.846·47-s + 0.142·49-s − 1.79·51-s − 0.316·53-s − 0.324·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.845150087\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.845150087\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - 7T \) |
| 11 | \( 1 + 11T \) |
| good | 3 | \( 1 + 5.83T + 27T^{2} \) |
| 5 | \( 1 - 12.0T + 125T^{2} \) |
| 13 | \( 1 - 31.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 112.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 6.19T + 6.85e3T^{2} \) |
| 23 | \( 1 + 113.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 71.8T + 2.43e4T^{2} \) |
| 31 | \( 1 - 88.9T + 2.97e4T^{2} \) |
| 37 | \( 1 + 92.9T + 5.06e4T^{2} \) |
| 41 | \( 1 - 192.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 124.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 272.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 122.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 67.4T + 2.05e5T^{2} \) |
| 61 | \( 1 + 173.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 22.9T + 3.00e5T^{2} \) |
| 71 | \( 1 + 769.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 57.1T + 3.89e5T^{2} \) |
| 79 | \( 1 - 381.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 177.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 243.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.81e3T + 9.12e5T^{2} \) |
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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
−9.582633210942833761928313381831, −8.505878694634617905107334942936, −7.69304162078894132946912393934, −6.55078937621559271922559234614, −5.75974940957905866312278343162, −5.50639898964938936734927263108, −4.39981572452100776390832556960, −3.06161202067350457889291945722, −1.76735724971766682439455203020, −0.76264664833339670601598011321,
0.76264664833339670601598011321, 1.76735724971766682439455203020, 3.06161202067350457889291945722, 4.39981572452100776390832556960, 5.50639898964938936734927263108, 5.75974940957905866312278343162, 6.55078937621559271922559234614, 7.69304162078894132946912393934, 8.505878694634617905107334942936, 9.582633210942833761928313381831
