| L(s) = 1 | + 2·3-s + 4·5-s − 7-s + 9-s + 8·15-s − 2·17-s + 2·19-s − 2·21-s + 8·23-s + 11·25-s − 4·27-s + 2·29-s − 4·31-s − 4·35-s + 6·37-s + 2·41-s + 8·43-s + 4·45-s + 4·47-s + 49-s − 4·51-s − 10·53-s + 4·57-s − 6·59-s + 4·61-s − 63-s + 12·67-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1.78·5-s − 0.377·7-s + 1/3·9-s + 2.06·15-s − 0.485·17-s + 0.458·19-s − 0.436·21-s + 1.66·23-s + 11/5·25-s − 0.769·27-s + 0.371·29-s − 0.718·31-s − 0.676·35-s + 0.986·37-s + 0.312·41-s + 1.21·43-s + 0.596·45-s + 0.583·47-s + 1/7·49-s − 0.560·51-s − 1.37·53-s + 0.529·57-s − 0.781·59-s + 0.512·61-s − 0.125·63-s + 1.46·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9464 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.732797505\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.732797505\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.72037261210953884219032595761, −6.98534613022119717882608766544, −6.36585694766889181183394644617, −5.65387948299863354531802329305, −5.07784839853149550370228376546, −4.10342578548114452648493428142, −3.05650399234090685875153001924, −2.67316607408775338708471856279, −1.95448976038181399591604434105, −1.02144060718188313014588525047,
1.02144060718188313014588525047, 1.95448976038181399591604434105, 2.67316607408775338708471856279, 3.05650399234090685875153001924, 4.10342578548114452648493428142, 5.07784839853149550370228376546, 5.65387948299863354531802329305, 6.36585694766889181183394644617, 6.98534613022119717882608766544, 7.72037261210953884219032595761
