| L(s) = 1 | + 0.319·2-s − 1.89·4-s − 1.26·5-s + 1.17·7-s − 1.24·8-s − 0.404·10-s − 1.82·11-s − 2.29·13-s + 0.375·14-s + 3.39·16-s − 1.44·17-s + 6.82·19-s + 2.40·20-s − 0.582·22-s − 23-s − 3.39·25-s − 0.734·26-s − 2.22·28-s + 29-s − 0.0171·31-s + 3.58·32-s − 0.461·34-s − 1.48·35-s + 11.3·37-s + 2.18·38-s + 1.57·40-s + 1.72·41-s + ⋯ |
| L(s) = 1 | + 0.226·2-s − 0.948·4-s − 0.565·5-s + 0.443·7-s − 0.440·8-s − 0.127·10-s − 0.549·11-s − 0.636·13-s + 0.100·14-s + 0.849·16-s − 0.350·17-s + 1.56·19-s + 0.536·20-s − 0.124·22-s − 0.208·23-s − 0.679·25-s − 0.144·26-s − 0.421·28-s + 0.185·29-s − 0.00307·31-s + 0.632·32-s − 0.0792·34-s − 0.251·35-s + 1.85·37-s + 0.354·38-s + 0.249·40-s + 0.269·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 - 0.319T + 2T^{2} \) |
| 5 | \( 1 + 1.26T + 5T^{2} \) |
| 7 | \( 1 - 1.17T + 7T^{2} \) |
| 11 | \( 1 + 1.82T + 11T^{2} \) |
| 13 | \( 1 + 2.29T + 13T^{2} \) |
| 17 | \( 1 + 1.44T + 17T^{2} \) |
| 19 | \( 1 - 6.82T + 19T^{2} \) |
| 31 | \( 1 + 0.0171T + 31T^{2} \) |
| 37 | \( 1 - 11.3T + 37T^{2} \) |
| 41 | \( 1 - 1.72T + 41T^{2} \) |
| 43 | \( 1 - 6.23T + 43T^{2} \) |
| 47 | \( 1 + 8.65T + 47T^{2} \) |
| 53 | \( 1 - 10.1T + 53T^{2} \) |
| 59 | \( 1 - 9.22T + 59T^{2} \) |
| 61 | \( 1 + 6.92T + 61T^{2} \) |
| 67 | \( 1 - 1.95T + 67T^{2} \) |
| 71 | \( 1 + 14.6T + 71T^{2} \) |
| 73 | \( 1 + 8.10T + 73T^{2} \) |
| 79 | \( 1 + 11.4T + 79T^{2} \) |
| 83 | \( 1 - 5.50T + 83T^{2} \) |
| 89 | \( 1 + 10.2T + 89T^{2} \) |
| 97 | \( 1 - 5.50T + 97T^{2} \) |
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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
−7.66591591495915800347117158669, −7.36089721109222934319422083240, −6.07748866501413332696257237183, −5.46624064342211266794071832764, −4.71569245513684991319862838204, −4.22834458860074446262836345403, −3.32471352289952087132543654359, −2.51266791695450579600360099795, −1.12616981014415666528100125315, 0,
1.12616981014415666528100125315, 2.51266791695450579600360099795, 3.32471352289952087132543654359, 4.22834458860074446262836345403, 4.71569245513684991319862838204, 5.46624064342211266794071832764, 6.07748866501413332696257237183, 7.36089721109222934319422083240, 7.66591591495915800347117158669
