| L(s) = 1 | − 3-s + 4·5-s − 3·7-s − 2·9-s + 2·11-s + 13-s − 4·15-s + 3·17-s − 19-s + 3·21-s + 23-s + 11·25-s + 5·27-s + 5·29-s + 8·31-s − 2·33-s − 12·35-s + 2·37-s − 39-s − 8·41-s + 4·43-s − 8·45-s − 8·47-s + 2·49-s − 3·51-s + 53-s + 8·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.78·5-s − 1.13·7-s − 2/3·9-s + 0.603·11-s + 0.277·13-s − 1.03·15-s + 0.727·17-s − 0.229·19-s + 0.654·21-s + 0.208·23-s + 11/5·25-s + 0.962·27-s + 0.928·29-s + 1.43·31-s − 0.348·33-s − 2.02·35-s + 0.328·37-s − 0.160·39-s − 1.24·41-s + 0.609·43-s − 1.19·45-s − 1.16·47-s + 2/7·49-s − 0.420·51-s + 0.137·53-s + 1.07·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.665917004\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.665917004\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - 4 T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 - 15 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 - 9 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−9.958642910212366556628274508824, −9.089395378892366384411617506988, −8.336890921646432967363618675577, −6.67486851525401293685577568233, −6.41602681477419049736409749590, −5.70311644291868843897751049539, −4.89838836690724431377597203004, −3.35219753314436267860210658260, −2.45747388367035991238545338827, −1.03106032747078173831338248002,
1.03106032747078173831338248002, 2.45747388367035991238545338827, 3.35219753314436267860210658260, 4.89838836690724431377597203004, 5.70311644291868843897751049539, 6.41602681477419049736409749590, 6.67486851525401293685577568233, 8.336890921646432967363618675577, 9.089395378892366384411617506988, 9.958642910212366556628274508824
