| L(s) = 1 | + 6.16·3-s + 16.2·5-s + 7·7-s + 11.0·9-s + 6.46·11-s + 47.8·13-s + 100.·15-s + 99.0·17-s − 90.6·19-s + 43.1·21-s + 76.5·23-s + 140.·25-s − 98.4·27-s − 144.·29-s + 250.·31-s + 39.8·33-s + 113.·35-s − 286.·37-s + 295.·39-s − 41·41-s + 0.197·43-s + 179.·45-s + 127.·47-s + 49·49-s + 610.·51-s + 292.·53-s + 105.·55-s + ⋯ |
| L(s) = 1 | + 1.18·3-s + 1.45·5-s + 0.377·7-s + 0.408·9-s + 0.177·11-s + 1.02·13-s + 1.72·15-s + 1.41·17-s − 1.09·19-s + 0.448·21-s + 0.694·23-s + 1.12·25-s − 0.701·27-s − 0.924·29-s + 1.45·31-s + 0.210·33-s + 0.550·35-s − 1.27·37-s + 1.21·39-s − 0.156·41-s + 0.000702·43-s + 0.595·45-s + 0.396·47-s + 0.142·49-s + 1.67·51-s + 0.757·53-s + 0.258·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.039979725\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.039979725\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - 7T \) |
| 41 | \( 1 + 41T \) |
| good | 3 | \( 1 - 6.16T + 27T^{2} \) |
| 5 | \( 1 - 16.2T + 125T^{2} \) |
| 11 | \( 1 - 6.46T + 1.33e3T^{2} \) |
| 13 | \( 1 - 47.8T + 2.19e3T^{2} \) |
| 17 | \( 1 - 99.0T + 4.91e3T^{2} \) |
| 19 | \( 1 + 90.6T + 6.85e3T^{2} \) |
| 23 | \( 1 - 76.5T + 1.21e4T^{2} \) |
| 29 | \( 1 + 144.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 250.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 286.T + 5.06e4T^{2} \) |
| 43 | \( 1 - 0.197T + 7.95e4T^{2} \) |
| 47 | \( 1 - 127.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 292.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 441.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 97.8T + 2.26e5T^{2} \) |
| 67 | \( 1 - 287.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 632.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 88.7T + 3.89e5T^{2} \) |
| 79 | \( 1 + 440.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.04e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 168.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 412.T + 9.12e5T^{2} \) |
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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
−9.300030145672322135898794692959, −8.666055230516543914607278695427, −8.070035438806370197712522998313, −6.94757283842951528973361483304, −5.99013728416437891734253786958, −5.31042497528525472854533201795, −3.96788894178595467846874044233, −3.01181235815204800862714931230, −2.07338540108645871700712511286, −1.23682253883057946217151359617,
1.23682253883057946217151359617, 2.07338540108645871700712511286, 3.01181235815204800862714931230, 3.96788894178595467846874044233, 5.31042497528525472854533201795, 5.99013728416437891734253786958, 6.94757283842951528973361483304, 8.070035438806370197712522998313, 8.666055230516543914607278695427, 9.300030145672322135898794692959
