| L(s) = 1 | − 8.89·3-s − 20.4·7-s + 52.1·9-s + 61.3·11-s + 45.1·13-s + 115.·17-s + 64.8·19-s + 182.·21-s + 6.11·23-s − 224.·27-s + 224.·29-s − 58.6·31-s − 546.·33-s − 99.6·37-s − 402.·39-s − 145.·41-s + 6.73·43-s − 203.·47-s + 77.0·49-s − 1.02e3·51-s + 275.·53-s − 576.·57-s + 262.·59-s + 790.·61-s − 1.06e3·63-s + 141.·67-s − 54.3·69-s + ⋯ |
| L(s) = 1 | − 1.71·3-s − 1.10·7-s + 1.93·9-s + 1.68·11-s + 0.964·13-s + 1.64·17-s + 0.782·19-s + 1.89·21-s + 0.0554·23-s − 1.59·27-s + 1.43·29-s − 0.339·31-s − 2.88·33-s − 0.442·37-s − 1.65·39-s − 0.555·41-s + 0.0238·43-s − 0.632·47-s + 0.224·49-s − 2.82·51-s + 0.714·53-s − 1.34·57-s + 0.578·59-s + 1.65·61-s − 2.13·63-s + 0.258·67-s − 0.0948·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.374323627\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.374323627\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 8.89T + 27T^{2} \) |
| 7 | \( 1 + 20.4T + 343T^{2} \) |
| 11 | \( 1 - 61.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 45.1T + 2.19e3T^{2} \) |
| 17 | \( 1 - 115.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 64.8T + 6.85e3T^{2} \) |
| 23 | \( 1 - 6.11T + 1.21e4T^{2} \) |
| 29 | \( 1 - 224.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 58.6T + 2.97e4T^{2} \) |
| 37 | \( 1 + 99.6T + 5.06e4T^{2} \) |
| 41 | \( 1 + 145.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 6.73T + 7.95e4T^{2} \) |
| 47 | \( 1 + 203.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 275.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 262.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 790.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 141.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.04e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.05e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 826.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.02e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 154.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 414.T + 9.12e5T^{2} \) |
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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
−9.361676525452360863715699861776, −8.225037538247398252122304911982, −6.90475088005619177326663656430, −6.61196531611028679472161828312, −5.84932284697701292565617837476, −5.18872846058738648083904883409, −3.97681822181324584106259037093, −3.32921301484977247632121944044, −1.31867608350126136470801554983, −0.73161143712649929664901714177,
0.73161143712649929664901714177, 1.31867608350126136470801554983, 3.32921301484977247632121944044, 3.97681822181324584106259037093, 5.18872846058738648083904883409, 5.84932284697701292565617837476, 6.61196531611028679472161828312, 6.90475088005619177326663656430, 8.225037538247398252122304911982, 9.361676525452360863715699861776
