| L(s) = 1 | − 3.52·2-s + 4.41·4-s − 25.4·7-s + 12.6·8-s + 71.3·11-s + 51.3·13-s + 89.6·14-s − 79.8·16-s − 33.3·17-s + 113.·19-s − 251.·22-s + 81.9·23-s − 181.·26-s − 112.·28-s + 246.·29-s + 222.·31-s + 180.·32-s + 117.·34-s − 22.3·37-s − 399.·38-s + 434.·41-s + 236.·43-s + 315.·44-s − 288.·46-s − 107.·47-s + 303.·49-s + 226.·52-s + ⋯ |
| L(s) = 1 | − 1.24·2-s + 0.551·4-s − 1.37·7-s + 0.558·8-s + 1.95·11-s + 1.09·13-s + 1.71·14-s − 1.24·16-s − 0.475·17-s + 1.36·19-s − 2.43·22-s + 0.743·23-s − 1.36·26-s − 0.757·28-s + 1.58·29-s + 1.29·31-s + 0.995·32-s + 0.592·34-s − 0.0994·37-s − 1.70·38-s + 1.65·41-s + 0.839·43-s + 1.07·44-s − 0.925·46-s − 0.335·47-s + 0.885·49-s + 0.605·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.325552439\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.325552439\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 3.52T + 8T^{2} \) |
| 7 | \( 1 + 25.4T + 343T^{2} \) |
| 11 | \( 1 - 71.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 51.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 33.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 113.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 81.9T + 1.21e4T^{2} \) |
| 29 | \( 1 - 246.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 222.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 22.3T + 5.06e4T^{2} \) |
| 41 | \( 1 - 434.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 236.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 107.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 123.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 171.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 79.4T + 2.26e5T^{2} \) |
| 67 | \( 1 - 611.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 511.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 410.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 793.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 270.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 177.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 881.T + 9.12e5T^{2} \) |
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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
−9.003619716492601933384916793903, −8.287277606918628533122476557113, −7.21913882369614379778197492121, −6.57255469350118867666588062147, −6.08325337654430023173501611939, −4.57736738896825337167224442246, −3.73609028456032331827459981824, −2.80338874243119669189563638065, −1.23483689083143993658502549481, −0.789063432266332745340188888962,
0.789063432266332745340188888962, 1.23483689083143993658502549481, 2.80338874243119669189563638065, 3.73609028456032331827459981824, 4.57736738896825337167224442246, 6.08325337654430023173501611939, 6.57255469350118867666588062147, 7.21913882369614379778197492121, 8.287277606918628533122476557113, 9.003619716492601933384916793903
