| L(s) = 1 | − 2.32·2-s − 0.105·3-s + 3.41·4-s − 1.05·5-s + 0.245·6-s + 0.945·7-s − 3.28·8-s − 2.98·9-s + 2.45·10-s + 2.00·11-s − 0.360·12-s + 4.81·13-s − 2.19·14-s + 0.111·15-s + 0.821·16-s + 5.66·17-s + 6.95·18-s + 0.270·19-s − 3.60·20-s − 0.0998·21-s − 4.66·22-s + 2.04·23-s + 0.347·24-s − 3.88·25-s − 11.1·26-s + 0.632·27-s + 3.22·28-s + ⋯ |
| L(s) = 1 | − 1.64·2-s − 0.0610·3-s + 1.70·4-s − 0.471·5-s + 0.100·6-s + 0.357·7-s − 1.16·8-s − 0.996·9-s + 0.776·10-s + 0.604·11-s − 0.104·12-s + 1.33·13-s − 0.587·14-s + 0.0287·15-s + 0.205·16-s + 1.37·17-s + 1.63·18-s + 0.0620·19-s − 0.805·20-s − 0.0217·21-s − 0.994·22-s + 0.426·23-s + 0.0709·24-s − 0.777·25-s − 2.19·26-s + 0.121·27-s + 0.609·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 53 | \( 1 + T \) |
| 151 | \( 1 + T \) |
| good | 2 | \( 1 + 2.32T + 2T^{2} \) |
| 3 | \( 1 + 0.105T + 3T^{2} \) |
| 5 | \( 1 + 1.05T + 5T^{2} \) |
| 7 | \( 1 - 0.945T + 7T^{2} \) |
| 11 | \( 1 - 2.00T + 11T^{2} \) |
| 13 | \( 1 - 4.81T + 13T^{2} \) |
| 17 | \( 1 - 5.66T + 17T^{2} \) |
| 19 | \( 1 - 0.270T + 19T^{2} \) |
| 23 | \( 1 - 2.04T + 23T^{2} \) |
| 29 | \( 1 + 5.94T + 29T^{2} \) |
| 31 | \( 1 - 3.29T + 31T^{2} \) |
| 37 | \( 1 + 5.04T + 37T^{2} \) |
| 41 | \( 1 + 5.06T + 41T^{2} \) |
| 43 | \( 1 - 1.81T + 43T^{2} \) |
| 47 | \( 1 - 12.2T + 47T^{2} \) |
| 59 | \( 1 + 11.0T + 59T^{2} \) |
| 61 | \( 1 + 7.84T + 61T^{2} \) |
| 67 | \( 1 + 11.3T + 67T^{2} \) |
| 71 | \( 1 + 6.29T + 71T^{2} \) |
| 73 | \( 1 + 1.26T + 73T^{2} \) |
| 79 | \( 1 + 9.25T + 79T^{2} \) |
| 83 | \( 1 + 15.9T + 83T^{2} \) |
| 89 | \( 1 + 9.97T + 89T^{2} \) |
| 97 | \( 1 - 13.8T + 97T^{2} \) |
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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
−7.55092719084633549140050753845, −7.28466521651584378837347344585, −6.04173590329822131842844197222, −5.89853815069092700761507956620, −4.65933044204258362832450000893, −3.62467683348938348534606610278, −3.00753826505861909512093435323, −1.75074769606546346516345794180, −1.11986025991099271836674417718, 0,
1.11986025991099271836674417718, 1.75074769606546346516345794180, 3.00753826505861909512093435323, 3.62467683348938348534606610278, 4.65933044204258362832450000893, 5.89853815069092700761507956620, 6.04173590329822131842844197222, 7.28466521651584378837347344585, 7.55092719084633549140050753845
