| L(s) = 1 | − 3-s − 5·5-s + 18·7-s − 26·9-s + 32·11-s − 47·13-s + 5·15-s + 20·17-s − 36·19-s − 18·21-s + 23·23-s + 25·25-s + 53·27-s − 27·29-s + 33·31-s − 32·33-s − 90·35-s + 56·37-s + 47·39-s − 157·41-s − 18·43-s + 130·45-s − 65·47-s − 19·49-s − 20·51-s − 14·53-s − 160·55-s + ⋯ |
| L(s) = 1 | − 0.192·3-s − 0.447·5-s + 0.971·7-s − 0.962·9-s + 0.877·11-s − 1.00·13-s + 0.0860·15-s + 0.285·17-s − 0.434·19-s − 0.187·21-s + 0.208·23-s + 1/5·25-s + 0.377·27-s − 0.172·29-s + 0.191·31-s − 0.168·33-s − 0.434·35-s + 0.248·37-s + 0.192·39-s − 0.598·41-s − 0.0638·43-s + 0.430·45-s − 0.201·47-s − 0.0553·49-s − 0.0549·51-s − 0.0362·53-s − 0.392·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.634804625\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.634804625\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + p T \) |
| 23 | \( 1 - p T \) |
| good | 3 | \( 1 + T + p^{3} T^{2} \) |
| 7 | \( 1 - 18 T + p^{3} T^{2} \) |
| 11 | \( 1 - 32 T + p^{3} T^{2} \) |
| 13 | \( 1 + 47 T + p^{3} T^{2} \) |
| 17 | \( 1 - 20 T + p^{3} T^{2} \) |
| 19 | \( 1 + 36 T + p^{3} T^{2} \) |
| 29 | \( 1 + 27 T + p^{3} T^{2} \) |
| 31 | \( 1 - 33 T + p^{3} T^{2} \) |
| 37 | \( 1 - 56 T + p^{3} T^{2} \) |
| 41 | \( 1 + 157 T + p^{3} T^{2} \) |
| 43 | \( 1 + 18 T + p^{3} T^{2} \) |
| 47 | \( 1 + 65 T + p^{3} T^{2} \) |
| 53 | \( 1 + 14 T + p^{3} T^{2} \) |
| 59 | \( 1 - 744 T + p^{3} T^{2} \) |
| 61 | \( 1 - 552 T + p^{3} T^{2} \) |
| 67 | \( 1 - 156 T + p^{3} T^{2} \) |
| 71 | \( 1 + 699 T + p^{3} T^{2} \) |
| 73 | \( 1 + 609 T + p^{3} T^{2} \) |
| 79 | \( 1 - 644 T + p^{3} T^{2} \) |
| 83 | \( 1 + 512 T + p^{3} T^{2} \) |
| 89 | \( 1 + 102 T + p^{3} T^{2} \) |
| 97 | \( 1 - 578 T + p^{3} T^{2} \) |
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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
−8.711715399439271082924714936795, −8.207305984458938237744496099716, −7.35133613432943717671285732127, −6.56293113033778307342122411517, −5.55189174981265763105046147047, −4.84944742656795121901203458479, −4.00098067263933781841643710759, −2.91335101440730798956477458132, −1.84052765001827419424482733174, −0.60247372986558779219696542568,
0.60247372986558779219696542568, 1.84052765001827419424482733174, 2.91335101440730798956477458132, 4.00098067263933781841643710759, 4.84944742656795121901203458479, 5.55189174981265763105046147047, 6.56293113033778307342122411517, 7.35133613432943717671285732127, 8.207305984458938237744496099716, 8.711715399439271082924714936795
