| L(s) = 1 | − 2·2-s + 2·4-s + 5-s − 7-s − 2·10-s + 4·11-s − 13-s + 2·14-s − 4·16-s − 2·17-s − 6·19-s + 2·20-s − 8·22-s − 3·23-s + 25-s + 2·26-s − 2·28-s − 2·29-s + 2·31-s + 8·32-s + 4·34-s − 35-s − 11·37-s + 12·38-s + 9·41-s − 8·43-s + 8·44-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s + 0.447·5-s − 0.377·7-s − 0.632·10-s + 1.20·11-s − 0.277·13-s + 0.534·14-s − 16-s − 0.485·17-s − 1.37·19-s + 0.447·20-s − 1.70·22-s − 0.625·23-s + 1/5·25-s + 0.392·26-s − 0.377·28-s − 0.371·29-s + 0.359·31-s + 1.41·32-s + 0.685·34-s − 0.169·35-s − 1.80·37-s + 1.94·38-s + 1.40·41-s − 1.21·43-s + 1.20·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1755 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1755 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 + p T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 11 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 + 15 T + p T^{2} \) |
| 79 | \( 1 - 15 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 - 7 T + p T^{2} \) |
| 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
−8.899137653826481787432460068661, −8.469698809173566475886580760296, −7.41076139649596452941106681533, −6.66527242573658373664250870453, −6.09810117582021041916721931905, −4.73027586525017639880436160479, −3.80198106531412178509459060570, −2.34324301240368937025222335170, −1.48008442760589146756179369117, 0,
1.48008442760589146756179369117, 2.34324301240368937025222335170, 3.80198106531412178509459060570, 4.73027586525017639880436160479, 6.09810117582021041916721931905, 6.66527242573658373664250870453, 7.41076139649596452941106681533, 8.469698809173566475886580760296, 8.899137653826481787432460068661
