| L(s) = 1 | − 2·3-s − 4·5-s − 2·7-s + 9-s − 6·11-s − 13-s + 8·15-s − 8·17-s − 7·19-s + 4·21-s − 9·23-s + 11·25-s + 4·27-s − 6·29-s − 6·31-s + 12·33-s + 8·35-s − 4·37-s + 2·39-s − 2·41-s + 5·43-s − 4·45-s − 3·49-s + 16·51-s − 6·53-s + 24·55-s + 14·57-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1.78·5-s − 0.755·7-s + 1/3·9-s − 1.80·11-s − 0.277·13-s + 2.06·15-s − 1.94·17-s − 1.60·19-s + 0.872·21-s − 1.87·23-s + 11/5·25-s + 0.769·27-s − 1.11·29-s − 1.07·31-s + 2.08·33-s + 1.35·35-s − 0.657·37-s + 0.320·39-s − 0.312·41-s + 0.762·43-s − 0.596·45-s − 3/7·49-s + 2.24·51-s − 0.824·53-s + 3.23·55-s + 1.85·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30376 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30376 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3797 | \( 1 + T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 8 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.92595014189507, −15.64234861513119, −15.08083110328672, −14.55843154946923, −13.58199547138613, −12.97073328959031, −12.67243219943437, −12.23389790931235, −11.59378100753876, −11.04161857036807, −10.76860725672937, −10.37686057515731, −9.507890997338932, −8.595562855046562, −8.384022462641684, −7.573758829419918, −7.201200827403869, −6.487796721061515, −5.979993841803799, −5.310543461569214, −4.476114756748991, −4.303826073777747, −3.465613178018053, −2.663208327258735, −1.938045659864408, 0, 0, 0,
1.938045659864408, 2.663208327258735, 3.465613178018053, 4.303826073777747, 4.476114756748991, 5.310543461569214, 5.979993841803799, 6.487796721061515, 7.201200827403869, 7.573758829419918, 8.384022462641684, 8.595562855046562, 9.507890997338932, 10.37686057515731, 10.76860725672937, 11.04161857036807, 11.59378100753876, 12.23389790931235, 12.67243219943437, 12.97073328959031, 13.58199547138613, 14.55843154946923, 15.08083110328672, 15.64234861513119, 15.92595014189507
