L(s) = 1 | − 2·4-s − 2·5-s + 2·7-s + 11-s + 5·13-s + 4·16-s + 3·17-s + 4·20-s − 6·23-s − 25-s − 4·28-s + 10·29-s + 10·31-s − 4·35-s + 10·41-s − 6·43-s − 2·44-s + 2·47-s − 3·49-s − 10·52-s − 5·53-s − 2·55-s + 5·59-s − 8·64-s − 10·65-s + 10·67-s − 6·68-s + ⋯ |
L(s) = 1 | − 4-s − 0.894·5-s + 0.755·7-s + 0.301·11-s + 1.38·13-s + 16-s + 0.727·17-s + 0.894·20-s − 1.25·23-s − 1/5·25-s − 0.755·28-s + 1.85·29-s + 1.79·31-s − 0.676·35-s + 1.56·41-s − 0.914·43-s − 0.301·44-s + 0.291·47-s − 3/7·49-s − 1.38·52-s − 0.686·53-s − 0.269·55-s + 0.650·59-s − 64-s − 1.24·65-s + 1.22·67-s − 0.727·68-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35739 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35739 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.195914830\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.195914830\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 - 15 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 - 5 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−14.90113894035141, −14.19036695123450, −13.87529323571655, −13.67793043359234, −12.61300981261088, −12.41201934984530, −11.70022466657102, −11.38642806180634, −10.69931737977166, −10.04650724535692, −9.688544064474393, −8.823526635085885, −8.370023618927428, −7.909071130584404, −7.815888638699164, −6.460534121422247, −6.312902925016955, −5.366486609057036, −4.808557343967551, −4.218175674708734, −3.769386447509718, −3.216960718641765, −2.157008198031361, −1.076597886545584, −0.7144410310207735,
0.7144410310207735, 1.076597886545584, 2.157008198031361, 3.216960718641765, 3.769386447509718, 4.218175674708734, 4.808557343967551, 5.366486609057036, 6.312902925016955, 6.460534121422247, 7.815888638699164, 7.909071130584404, 8.370023618927428, 8.823526635085885, 9.688544064474393, 10.04650724535692, 10.69931737977166, 11.38642806180634, 11.70022466657102, 12.41201934984530, 12.61300981261088, 13.67793043359234, 13.87529323571655, 14.19036695123450, 14.90113894035141