| L(s) = 1 | + 3-s + 5-s − 3·7-s + 9-s + 2·11-s − 5·13-s + 15-s + 2·17-s − 19-s − 3·21-s + 2·23-s + 25-s + 27-s − 5·29-s + 4·31-s + 2·33-s − 3·35-s − 5·39-s + 10·41-s + 6·43-s + 45-s + 12·47-s + 2·49-s + 2·51-s + 6·53-s + 2·55-s − 57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 1.13·7-s + 1/3·9-s + 0.603·11-s − 1.38·13-s + 0.258·15-s + 0.485·17-s − 0.229·19-s − 0.654·21-s + 0.417·23-s + 1/5·25-s + 0.192·27-s − 0.928·29-s + 0.718·31-s + 0.348·33-s − 0.507·35-s − 0.800·39-s + 1.56·41-s + 0.914·43-s + 0.149·45-s + 1.75·47-s + 2/7·49-s + 0.280·51-s + 0.824·53-s + 0.269·55-s − 0.132·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 328560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 328560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.329707594\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.329707594\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 37 | \( 1 \) |
| good | 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 13 T + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 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
−12.72741953059768, −12.24366514608743, −11.89360084166237, −11.20443112087202, −10.62314757341592, −10.23564607515954, −9.793139294595735, −9.308044443185269, −9.135209636056523, −8.709678899858338, −7.779880037560665, −7.526587744469960, −7.160375738082559, −6.511732851457640, −6.082391968218531, −5.701039086131736, −4.985990333578523, −4.477266620834771, −3.953451800165147, −3.385314309871609, −2.874778714074048, −2.388881095817125, −1.956569503259513, −1.019587536106739, −0.5103380000604765,
0.5103380000604765, 1.019587536106739, 1.956569503259513, 2.388881095817125, 2.874778714074048, 3.385314309871609, 3.953451800165147, 4.477266620834771, 4.985990333578523, 5.701039086131736, 6.082391968218531, 6.511732851457640, 7.160375738082559, 7.526587744469960, 7.779880037560665, 8.709678899858338, 9.135209636056523, 9.308044443185269, 9.793139294595735, 10.23564607515954, 10.62314757341592, 11.20443112087202, 11.89360084166237, 12.24366514608743, 12.72741953059768
