| L(s) = 1 | + 2.81·2-s − 0.0599·4-s − 29.9·7-s − 22.7·8-s + 14.3·11-s − 45.4·13-s − 84.3·14-s − 63.5·16-s + 90.4·17-s − 30.5·19-s + 40.3·22-s + 134.·23-s − 128.·26-s + 1.79·28-s − 155.·29-s − 175.·31-s + 2.71·32-s + 254.·34-s − 161.·37-s − 86.0·38-s + 274.·41-s + 326.·43-s − 0.857·44-s + 379.·46-s + 476.·47-s + 552.·49-s + 2.72·52-s + ⋯ |
| L(s) = 1 | + 0.996·2-s − 0.00749·4-s − 1.61·7-s − 1.00·8-s + 0.392·11-s − 0.969·13-s − 1.60·14-s − 0.992·16-s + 1.29·17-s − 0.368·19-s + 0.390·22-s + 1.22·23-s − 0.965·26-s + 0.0121·28-s − 0.995·29-s − 1.01·31-s + 0.0149·32-s + 1.28·34-s − 0.716·37-s − 0.367·38-s + 1.04·41-s + 1.15·43-s − 0.00293·44-s + 1.21·46-s + 1.47·47-s + 1.60·49-s + 0.00726·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1125 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1125 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.851769444\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.851769444\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 2.81T + 8T^{2} \) |
| 7 | \( 1 + 29.9T + 343T^{2} \) |
| 11 | \( 1 - 14.3T + 1.33e3T^{2} \) |
| 13 | \( 1 + 45.4T + 2.19e3T^{2} \) |
| 17 | \( 1 - 90.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 30.5T + 6.85e3T^{2} \) |
| 23 | \( 1 - 134.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 155.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 175.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 161.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 274.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 326.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 476.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 103.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 442.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 730.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 1.00e3T + 3.00e5T^{2} \) |
| 71 | \( 1 - 540.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 257.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.01e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.10e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 240.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.51e3T + 9.12e5T^{2} \) |
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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
−9.404803879444178366493302458777, −8.955304790476912641820048446859, −7.49294110401847821249081588259, −6.79385734206848241006292344106, −5.84219072445847739079527194250, −5.26878156839766383212222384503, −4.04354553272282027780390178112, −3.39422531677602852334685387332, −2.52937573360639100553614100088, −0.58512361932278925958204551616,
0.58512361932278925958204551616, 2.52937573360639100553614100088, 3.39422531677602852334685387332, 4.04354553272282027780390178112, 5.26878156839766383212222384503, 5.84219072445847739079527194250, 6.79385734206848241006292344106, 7.49294110401847821249081588259, 8.955304790476912641820048446859, 9.404803879444178366493302458777
