| L(s) = 1 | − 11·5-s − 19·7-s − 38·11-s − 13·13-s + 51·17-s − 90·19-s − 52·23-s − 4·25-s + 190·29-s − 292·31-s + 209·35-s − 441·37-s − 312·41-s − 373·43-s − 41·47-s + 18·49-s − 468·53-s + 418·55-s + 530·59-s + 592·61-s + 143·65-s + 206·67-s − 863·71-s − 322·73-s + 722·77-s + 460·79-s + 528·83-s + ⋯ |
| L(s) = 1 | − 0.983·5-s − 1.02·7-s − 1.04·11-s − 0.277·13-s + 0.727·17-s − 1.08·19-s − 0.471·23-s − 0.0319·25-s + 1.21·29-s − 1.69·31-s + 1.00·35-s − 1.95·37-s − 1.18·41-s − 1.32·43-s − 0.127·47-s + 0.0524·49-s − 1.21·53-s + 1.02·55-s + 1.16·59-s + 1.24·61-s + 0.272·65-s + 0.375·67-s − 1.44·71-s − 0.516·73-s + 1.06·77-s + 0.655·79-s + 0.698·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.2153743129\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2153743129\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 + p T \) |
| good | 5 | \( 1 + 11 T + p^{3} T^{2} \) |
| 7 | \( 1 + 19 T + p^{3} T^{2} \) |
| 11 | \( 1 + 38 T + p^{3} T^{2} \) |
| 17 | \( 1 - 3 p T + p^{3} T^{2} \) |
| 19 | \( 1 + 90 T + p^{3} T^{2} \) |
| 23 | \( 1 + 52 T + p^{3} T^{2} \) |
| 29 | \( 1 - 190 T + p^{3} T^{2} \) |
| 31 | \( 1 + 292 T + p^{3} T^{2} \) |
| 37 | \( 1 + 441 T + p^{3} T^{2} \) |
| 41 | \( 1 + 312 T + p^{3} T^{2} \) |
| 43 | \( 1 + 373 T + p^{3} T^{2} \) |
| 47 | \( 1 + 41 T + p^{3} T^{2} \) |
| 53 | \( 1 + 468 T + p^{3} T^{2} \) |
| 59 | \( 1 - 530 T + p^{3} T^{2} \) |
| 61 | \( 1 - 592 T + p^{3} T^{2} \) |
| 67 | \( 1 - 206 T + p^{3} T^{2} \) |
| 71 | \( 1 + 863 T + p^{3} T^{2} \) |
| 73 | \( 1 + 322 T + p^{3} T^{2} \) |
| 79 | \( 1 - 460 T + p^{3} T^{2} \) |
| 83 | \( 1 - 528 T + p^{3} T^{2} \) |
| 89 | \( 1 + 870 T + p^{3} T^{2} \) |
| 97 | \( 1 + 346 T + p^{3} T^{2} \) |
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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
−8.633084731560634841762541678578, −8.185821482711142429506878648218, −7.25946498792853728537714450575, −6.66264714842955144518649957343, −5.61833531076953338593506480530, −4.79612357784681838862234912461, −3.68817818207649736366286025180, −3.15994272406812893275633752816, −1.93390172542120661408955576717, −0.20342271699193499527119311717,
0.20342271699193499527119311717, 1.93390172542120661408955576717, 3.15994272406812893275633752816, 3.68817818207649736366286025180, 4.79612357784681838862234912461, 5.61833531076953338593506480530, 6.66264714842955144518649957343, 7.25946498792853728537714450575, 8.185821482711142429506878648218, 8.633084731560634841762541678578
