| L(s) = 1 | + 11.8·5-s − 9.85·7-s + 39.0·11-s + 91.5·13-s + 37.1·17-s + 46.4·19-s − 120.·23-s + 15.5·25-s + 27.2·29-s + 81.1·31-s − 116.·35-s + 10.9·37-s + 205.·41-s − 115.·43-s − 312.·47-s − 245.·49-s − 90.9·53-s + 463.·55-s − 550.·59-s + 630.·61-s + 1.08e3·65-s + 661.·67-s + 494.·71-s + 566.·73-s − 385.·77-s − 49.4·79-s + 564.·83-s + ⋯ |
| L(s) = 1 | + 1.06·5-s − 0.532·7-s + 1.07·11-s + 1.95·13-s + 0.529·17-s + 0.561·19-s − 1.09·23-s + 0.124·25-s + 0.174·29-s + 0.470·31-s − 0.564·35-s + 0.0488·37-s + 0.782·41-s − 0.409·43-s − 0.970·47-s − 0.716·49-s − 0.235·53-s + 1.13·55-s − 1.21·59-s + 1.32·61-s + 2.07·65-s + 1.20·67-s + 0.827·71-s + 0.907·73-s − 0.569·77-s − 0.0703·79-s + 0.745·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.082115655\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.082115655\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 11.8T + 125T^{2} \) |
| 7 | \( 1 + 9.85T + 343T^{2} \) |
| 11 | \( 1 - 39.0T + 1.33e3T^{2} \) |
| 13 | \( 1 - 91.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 37.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 46.4T + 6.85e3T^{2} \) |
| 23 | \( 1 + 120.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 27.2T + 2.43e4T^{2} \) |
| 31 | \( 1 - 81.1T + 2.97e4T^{2} \) |
| 37 | \( 1 - 10.9T + 5.06e4T^{2} \) |
| 41 | \( 1 - 205.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 115.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 312.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 90.9T + 1.48e5T^{2} \) |
| 59 | \( 1 + 550.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 630.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 661.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 494.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 566.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 49.4T + 4.93e5T^{2} \) |
| 83 | \( 1 - 564.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.08e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 464.T + 9.12e5T^{2} \) |
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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
−9.573246124929127709405737165841, −8.711883612088204410309204229018, −7.900795239583206496533982475286, −6.45662512772631035007470417003, −6.29489710607757447726469706472, −5.37733307028289133126446210314, −4.00351961040418126977313039840, −3.27090056278619453963928088953, −1.86172203544796204926168754380, −0.983132755291107895015842329883,
0.983132755291107895015842329883, 1.86172203544796204926168754380, 3.27090056278619453963928088953, 4.00351961040418126977313039840, 5.37733307028289133126446210314, 6.29489710607757447726469706472, 6.45662512772631035007470417003, 7.900795239583206496533982475286, 8.711883612088204410309204229018, 9.573246124929127709405737165841
