| L(s) = 1 | − 1.43·2-s − 5.93·4-s + 7·7-s + 20.0·8-s − 7.61·11-s − 52.3·13-s − 10.0·14-s + 18.6·16-s − 49.7·17-s + 140.·19-s + 10.9·22-s − 23.4·23-s + 75.3·26-s − 41.5·28-s − 157.·29-s + 127.·31-s − 187.·32-s + 71.5·34-s + 115.·37-s − 202.·38-s + 188.·41-s − 322.·43-s + 45.1·44-s + 33.7·46-s − 76.6·47-s + 49·49-s + 310.·52-s + ⋯ |
| L(s) = 1 | − 0.508·2-s − 0.741·4-s + 0.377·7-s + 0.885·8-s − 0.208·11-s − 1.11·13-s − 0.192·14-s + 0.290·16-s − 0.709·17-s + 1.69·19-s + 0.106·22-s − 0.212·23-s + 0.568·26-s − 0.280·28-s − 1.00·29-s + 0.740·31-s − 1.03·32-s + 0.360·34-s + 0.513·37-s − 0.863·38-s + 0.718·41-s − 1.14·43-s + 0.154·44-s + 0.108·46-s − 0.237·47-s + 0.142·49-s + 0.828·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.000082946\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.000082946\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - 7T \) |
| good | 2 | \( 1 + 1.43T + 8T^{2} \) |
| 11 | \( 1 + 7.61T + 1.33e3T^{2} \) |
| 13 | \( 1 + 52.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 49.7T + 4.91e3T^{2} \) |
| 19 | \( 1 - 140.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 23.4T + 1.21e4T^{2} \) |
| 29 | \( 1 + 157.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 127.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 115.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 188.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 322.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 76.6T + 1.03e5T^{2} \) |
| 53 | \( 1 + 424.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 107.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 915.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 451.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 907.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 755.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 22.5T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.11e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.51e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 549.T + 9.12e5T^{2} \) |
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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
−9.223212462190815399640567056115, −8.213209267102001433560140016450, −7.67161160038063862620248009178, −6.91085803112935730018740208669, −5.57636034489855508173438237324, −4.93592174007107908795049562490, −4.14130297391120576408425657680, −2.94106904635594231010568908426, −1.71130330139295447440015412653, −0.53501053159481024997518667391,
0.53501053159481024997518667391, 1.71130330139295447440015412653, 2.94106904635594231010568908426, 4.14130297391120576408425657680, 4.93592174007107908795049562490, 5.57636034489855508173438237324, 6.91085803112935730018740208669, 7.67161160038063862620248009178, 8.213209267102001433560140016450, 9.223212462190815399640567056115
