| L(s) = 1 | − 2·2-s − 2·3-s + 4·4-s + 5·5-s + 4·6-s + 8·7-s − 8·8-s − 23·9-s − 10·10-s + 44·11-s − 8·12-s − 16·14-s − 10·15-s + 16·16-s − 74·17-s + 46·18-s + 19·19-s + 20·20-s − 16·21-s − 88·22-s + 84·23-s + 16·24-s + 25·25-s + 100·27-s + 32·28-s + 266·29-s + 20·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.384·3-s + 1/2·4-s + 0.447·5-s + 0.272·6-s + 0.431·7-s − 0.353·8-s − 0.851·9-s − 0.316·10-s + 1.20·11-s − 0.192·12-s − 0.305·14-s − 0.172·15-s + 1/4·16-s − 1.05·17-s + 0.602·18-s + 0.229·19-s + 0.223·20-s − 0.166·21-s − 0.852·22-s + 0.761·23-s + 0.136·24-s + 1/5·25-s + 0.712·27-s + 0.215·28-s + 1.70·29-s + 0.121·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 190 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.215008403\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.215008403\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p T \) |
| 5 | \( 1 - p T \) |
| 19 | \( 1 - p T \) |
| good | 3 | \( 1 + 2 T + p^{3} T^{2} \) |
| 7 | \( 1 - 8 T + p^{3} T^{2} \) |
| 11 | \( 1 - 4 p T + p^{3} T^{2} \) |
| 13 | \( 1 + p^{3} T^{2} \) |
| 17 | \( 1 + 74 T + p^{3} T^{2} \) |
| 23 | \( 1 - 84 T + p^{3} T^{2} \) |
| 29 | \( 1 - 266 T + p^{3} T^{2} \) |
| 31 | \( 1 - 136 T + p^{3} T^{2} \) |
| 37 | \( 1 - 424 T + p^{3} T^{2} \) |
| 41 | \( 1 - 470 T + p^{3} T^{2} \) |
| 43 | \( 1 + 236 T + p^{3} T^{2} \) |
| 47 | \( 1 + 240 T + p^{3} T^{2} \) |
| 53 | \( 1 - 36 T + p^{3} T^{2} \) |
| 59 | \( 1 - 736 T + p^{3} T^{2} \) |
| 61 | \( 1 - 650 T + p^{3} T^{2} \) |
| 67 | \( 1 + 830 T + p^{3} T^{2} \) |
| 71 | \( 1 + 216 T + p^{3} T^{2} \) |
| 73 | \( 1 - 254 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1220 T + p^{3} T^{2} \) |
| 83 | \( 1 + 688 T + p^{3} T^{2} \) |
| 89 | \( 1 - 102 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1280 T + p^{3} T^{2} \) |
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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
−11.61804268974101882712418321278, −11.30108786210578150047726487526, −10.06563248919596086927317763643, −9.045808992804952527998531857065, −8.294683239727992791054403451043, −6.80864540432929144700634262843, −6.03007862468296448155260033444, −4.59130582116488867936543033168, −2.67419705968290949548215411095, −1.00997100961400233228225279044,
1.00997100961400233228225279044, 2.67419705968290949548215411095, 4.59130582116488867936543033168, 6.03007862468296448155260033444, 6.80864540432929144700634262843, 8.294683239727992791054403451043, 9.045808992804952527998531857065, 10.06563248919596086927317763643, 11.30108786210578150047726487526, 11.61804268974101882712418321278
