L(s) = 1 | − 3-s + 2·5-s − 4·7-s + 9-s + 11-s + 4·13-s − 2·15-s − 17-s + 8·19-s + 4·21-s − 25-s − 27-s + 10·31-s − 33-s − 8·35-s − 8·37-s − 4·39-s − 10·41-s + 8·43-s + 2·45-s + 10·47-s + 9·49-s + 51-s + 12·53-s + 2·55-s − 8·57-s + 8·59-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.894·5-s − 1.51·7-s + 1/3·9-s + 0.301·11-s + 1.10·13-s − 0.516·15-s − 0.242·17-s + 1.83·19-s + 0.872·21-s − 1/5·25-s − 0.192·27-s + 1.79·31-s − 0.174·33-s − 1.35·35-s − 1.31·37-s − 0.640·39-s − 1.56·41-s + 1.21·43-s + 0.298·45-s + 1.45·47-s + 9/7·49-s + 0.140·51-s + 1.64·53-s + 0.269·55-s − 1.05·57-s + 1.04·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35904 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35904 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.274330552\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.274330552\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 17 | \( 1 + T \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−15.18875173686903, −13.96119991582028, −13.75960394034428, −13.55127753773240, −12.86336110520109, −12.23355666949998, −11.84734322870980, −11.30723322338989, −10.42922359943445, −10.15915131297985, −9.725311102792712, −9.086910321793559, −8.719839483037692, −7.822447159919679, −7.001883317340722, −6.685271427624544, −6.116945692489470, −5.603642392976728, −5.216068322502588, −4.153543588661546, −3.605720269080526, −2.991532108661733, −2.227778041680866, −1.231999039357755, −0.6521504101355587,
0.6521504101355587, 1.231999039357755, 2.227778041680866, 2.991532108661733, 3.605720269080526, 4.153543588661546, 5.216068322502588, 5.603642392976728, 6.116945692489470, 6.685271427624544, 7.001883317340722, 7.822447159919679, 8.719839483037692, 9.086910321793559, 9.725311102792712, 10.15915131297985, 10.42922359943445, 11.30723322338989, 11.84734322870980, 12.23355666949998, 12.86336110520109, 13.55127753773240, 13.75960394034428, 13.96119991582028, 15.18875173686903