L(s) = 1 | + 2·5-s + 2·13-s + 6·17-s + 4·23-s − 25-s − 2·29-s + 10·37-s + 6·41-s + 8·43-s − 4·47-s − 7·49-s − 6·53-s + 12·59-s + 2·61-s + 4·65-s + 4·67-s + 12·71-s + 14·73-s + 16·79-s − 12·83-s + 12·85-s − 10·89-s − 14·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
L(s) = 1 | + 0.894·5-s + 0.554·13-s + 1.45·17-s + 0.834·23-s − 1/5·25-s − 0.371·29-s + 1.64·37-s + 0.937·41-s + 1.21·43-s − 0.583·47-s − 49-s − 0.824·53-s + 1.56·59-s + 0.256·61-s + 0.496·65-s + 0.488·67-s + 1.42·71-s + 1.63·73-s + 1.80·79-s − 1.31·83-s + 1.30·85-s − 1.05·89-s − 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69696 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69696 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.033847697\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.033847697\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−14.06909765916135, −13.74637145939397, −13.09637993728003, −12.66718086912565, −12.37488175260698, −11.43051446288444, −11.17177091966997, −10.68768505078604, −9.894066124157090, −9.530023237664598, −9.391173238752794, −8.391131647897418, −8.064464182397745, −7.488044816286560, −6.806106306591203, −6.258289777555888, −5.718521848725470, −5.374775543616987, −4.678249598887051, −3.918342158571638, −3.384342114218865, −2.660048369261690, −2.084463693772574, −1.235513134900441, −0.7443570595370874,
0.7443570595370874, 1.235513134900441, 2.084463693772574, 2.660048369261690, 3.384342114218865, 3.918342158571638, 4.678249598887051, 5.374775543616987, 5.718521848725470, 6.258289777555888, 6.806106306591203, 7.488044816286560, 8.064464182397745, 8.391131647897418, 9.391173238752794, 9.530023237664598, 9.894066124157090, 10.68768505078604, 11.17177091966997, 11.43051446288444, 12.37488175260698, 12.66718086912565, 13.09637993728003, 13.74637145939397, 14.06909765916135