L(s) = 1 | − 3-s + 5·7-s − 2·9-s − 5·11-s − 13-s − 4·17-s − 4·19-s − 5·21-s + 9·23-s − 5·25-s + 5·27-s − 2·29-s − 2·31-s + 5·33-s + 6·37-s + 39-s + 6·41-s − 5·43-s − 47-s + 18·49-s + 4·51-s + 6·53-s + 4·57-s − 12·59-s + 3·61-s − 10·63-s − 5·67-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1.88·7-s − 2/3·9-s − 1.50·11-s − 0.277·13-s − 0.970·17-s − 0.917·19-s − 1.09·21-s + 1.87·23-s − 25-s + 0.962·27-s − 0.371·29-s − 0.359·31-s + 0.870·33-s + 0.986·37-s + 0.160·39-s + 0.937·41-s − 0.762·43-s − 0.145·47-s + 18/7·49-s + 0.560·51-s + 0.824·53-s + 0.529·57-s − 1.56·59-s + 0.384·61-s − 1.25·63-s − 0.610·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 503 | \( 1 - T \) |
good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 5 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 3 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 7 T + p T^{2} \) |
| 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
−15.12849931885880, −14.90225489865164, −14.41972239212832, −13.61518669348707, −13.26536657524973, −12.69945185062002, −11.99321362398485, −11.41092119202366, −11.07129583215718, −10.75373355886036, −10.25689005091162, −9.221241407140362, −8.760750227884515, −8.249993552213817, −7.637850082266348, −7.330695536849705, −6.362658414291809, −5.763291660774942, −5.189644100580913, −4.752708205235244, −4.365197400753711, −3.217218847938887, −2.373917254477457, −2.020980104363411, −0.9159310831249025, 0,
0.9159310831249025, 2.020980104363411, 2.373917254477457, 3.217218847938887, 4.365197400753711, 4.752708205235244, 5.189644100580913, 5.763291660774942, 6.362658414291809, 7.330695536849705, 7.637850082266348, 8.249993552213817, 8.760750227884515, 9.221241407140362, 10.25689005091162, 10.75373355886036, 11.07129583215718, 11.41092119202366, 11.99321362398485, 12.69945185062002, 13.26536657524973, 13.61518669348707, 14.41972239212832, 14.90225489865164, 15.12849931885880