| L(s) = 1 | + 4·2-s − 12·3-s + 16·4-s − 48·6-s − 54·7-s + 64·8-s − 99·9-s − 121·11-s − 192·12-s + 540·13-s − 216·14-s + 256·16-s − 340·17-s − 396·18-s − 952·19-s + 648·21-s − 484·22-s − 1.09e3·23-s − 768·24-s + 2.16e3·26-s + 4.10e3·27-s − 864·28-s − 62·29-s − 7.56e3·31-s + 1.02e3·32-s + 1.45e3·33-s − 1.36e3·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.769·3-s + 1/2·4-s − 0.544·6-s − 0.416·7-s + 0.353·8-s − 0.407·9-s − 0.301·11-s − 0.384·12-s + 0.886·13-s − 0.294·14-s + 1/4·16-s − 0.285·17-s − 0.288·18-s − 0.604·19-s + 0.320·21-s − 0.213·22-s − 0.430·23-s − 0.272·24-s + 0.626·26-s + 1.08·27-s − 0.208·28-s − 0.0136·29-s − 1.41·31-s + 0.176·32-s + 0.232·33-s − 0.201·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 550 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.923659891\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.923659891\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{2} T \) |
| 5 | \( 1 \) |
| 11 | \( 1 + p^{2} T \) |
| good | 3 | \( 1 + 4 p T + p^{5} T^{2} \) |
| 7 | \( 1 + 54 T + p^{5} T^{2} \) |
| 13 | \( 1 - 540 T + p^{5} T^{2} \) |
| 17 | \( 1 + 20 p T + p^{5} T^{2} \) |
| 19 | \( 1 + 952 T + p^{5} T^{2} \) |
| 23 | \( 1 + 1092 T + p^{5} T^{2} \) |
| 29 | \( 1 + 62 T + p^{5} T^{2} \) |
| 31 | \( 1 + 7560 T + p^{5} T^{2} \) |
| 37 | \( 1 - 9186 T + p^{5} T^{2} \) |
| 41 | \( 1 + 6818 T + p^{5} T^{2} \) |
| 43 | \( 1 - 13310 T + p^{5} T^{2} \) |
| 47 | \( 1 - 22420 T + p^{5} T^{2} \) |
| 53 | \( 1 + 19654 T + p^{5} T^{2} \) |
| 59 | \( 1 - 48292 T + p^{5} T^{2} \) |
| 61 | \( 1 - 17530 T + p^{5} T^{2} \) |
| 67 | \( 1 - 35344 T + p^{5} T^{2} \) |
| 71 | \( 1 + 22912 T + p^{5} T^{2} \) |
| 73 | \( 1 + 47852 T + p^{5} T^{2} \) |
| 79 | \( 1 - 52396 T + p^{5} T^{2} \) |
| 83 | \( 1 + 7890 T + p^{5} T^{2} \) |
| 89 | \( 1 - 41958 T + p^{5} T^{2} \) |
| 97 | \( 1 - 37602 T + p^{5} T^{2} \) |
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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
−10.35058802351780721677457599000, −9.135264351073326913665845711425, −8.155616085305958331443813569918, −6.97685599497818512267969668630, −6.09346494683228218795199206524, −5.55536320263920430355280103960, −4.40485537499184903148190596113, −3.38853311615114792164802786861, −2.16899600376030820963736545494, −0.62141487168798045446959124604,
0.62141487168798045446959124604, 2.16899600376030820963736545494, 3.38853311615114792164802786861, 4.40485537499184903148190596113, 5.55536320263920430355280103960, 6.09346494683228218795199206524, 6.97685599497818512267969668630, 8.155616085305958331443813569918, 9.135264351073326913665845711425, 10.35058802351780721677457599000
