L(s) = 1 | − i·3-s + i·4-s − i·5-s − 9-s + i·11-s + 12-s − 15-s − 16-s + 20-s + (−1 + i)23-s − 25-s + i·27-s + 33-s − i·36-s + (1 − i)37-s + ⋯ |
L(s) = 1 | − i·3-s + i·4-s − i·5-s − 9-s + i·11-s + 12-s − 15-s − 16-s + 20-s + (−1 + i)23-s − 25-s + i·27-s + 33-s − i·36-s + (1 − i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6396858569\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6396858569\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + iT \) |
| 5 | \( 1 + iT \) |
| 11 | \( 1 - iT \) |
good | 2 | \( 1 - iT^{2} \) |
| 7 | \( 1 + iT^{2} \) |
| 13 | \( 1 - iT^{2} \) |
| 17 | \( 1 - iT^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + (1 - i)T - iT^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 + (-1 + i)T - iT^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 47 | \( 1 + (1 + i)T + iT^{2} \) |
| 53 | \( 1 + (-1 + i)T - iT^{2} \) |
| 59 | \( 1 - 2T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + (1 - i)T - iT^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - iT^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 + iT^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + (1 - i)T - iT^{2} \) |
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
−12.99948646994780214470491825403, −12.08663318021246051986637663717, −11.57875433332852445614781140229, −9.728772459251592395121557522403, −8.616384978317592918256578488262, −7.83096366481836823171257604871, −6.93281965314840913390216512087, −5.42246871145185096342319339715, −3.95015737579757376172805773740, −2.09761890952711296516119355548,
2.78769295406376910388561306770, 4.27941801431036193310475328141, 5.70988805836185351027648661991, 6.46389700689867982798864646199, 8.187516102217810345877015148760, 9.416118499399610651969579898083, 10.28540261466507186940224189090, 10.91510297876007671294013685694, 11.71565787232289332281999171781, 13.58387184974898777730455684846