| L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s − 8-s + 9-s + 10-s + 12-s − 6·13-s − 15-s + 16-s − 3·17-s − 18-s + 19-s − 20-s − 6·23-s − 24-s + 25-s + 6·26-s + 27-s + 6·29-s + 30-s + 31-s − 32-s + 3·34-s + 36-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.288·12-s − 1.66·13-s − 0.258·15-s + 1/4·16-s − 0.727·17-s − 0.235·18-s + 0.229·19-s − 0.223·20-s − 1.25·23-s − 0.204·24-s + 1/5·25-s + 1.17·26-s + 0.192·27-s + 1.11·29-s + 0.182·30-s + 0.179·31-s − 0.176·32-s + 0.514·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 68970 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 68970 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7328348148\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7328348148\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 11 T + p T^{2} \) |
| 59 | \( 1 + T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 15 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 7 T + p T^{2} \) |
| 89 | \( 1 - 5 T + p T^{2} \) |
| 97 | \( 1 - 5 T + p T^{2} \) |
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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
−14.30809247923776, −13.79977012663968, −12.98390957282110, −12.67430260198019, −12.10006667724086, −11.50026286727821, −11.31428913634356, −10.34464533911402, −10.01321409027974, −9.653775289929498, −9.074779834628269, −8.463678715096831, −7.883133002936385, −7.782470375038338, −6.931256167844068, −6.629281711887499, −5.926100327895830, −5.047773268355114, −4.571205067511891, −4.022144858392944, −3.146300949160328, −2.650557474065736, −2.107688738343478, −1.331858592836028, −0.3064091597008195,
0.3064091597008195, 1.331858592836028, 2.107688738343478, 2.650557474065736, 3.146300949160328, 4.022144858392944, 4.571205067511891, 5.047773268355114, 5.926100327895830, 6.629281711887499, 6.931256167844068, 7.782470375038338, 7.883133002936385, 8.463678715096831, 9.074779834628269, 9.653775289929498, 10.01321409027974, 10.34464533911402, 11.31428913634356, 11.50026286727821, 12.10006667724086, 12.67430260198019, 12.98390957282110, 13.79977012663968, 14.30809247923776
