| L(s) = 1 | + 2-s + 3-s + 4-s + 4.41·5-s + 6-s + 8-s + 9-s + 4.41·10-s − 11-s + 12-s + 2.82·13-s + 4.41·15-s + 16-s − 4.24·17-s + 18-s − 19-s + 4.41·20-s − 22-s + 0.585·23-s + 24-s + 14.4·25-s + 2.82·26-s + 27-s + 7.82·29-s + 4.41·30-s − 0.656·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.97·5-s + 0.408·6-s + 0.353·8-s + 0.333·9-s + 1.39·10-s − 0.301·11-s + 0.288·12-s + 0.784·13-s + 1.13·15-s + 0.250·16-s − 1.02·17-s + 0.235·18-s − 0.229·19-s + 0.987·20-s − 0.213·22-s + 0.122·23-s + 0.204·24-s + 2.89·25-s + 0.554·26-s + 0.192·27-s + 1.45·29-s + 0.805·30-s − 0.117·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5586 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5586 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.069700107\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.069700107\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 - 4.41T + 5T^{2} \) |
| 11 | \( 1 + T + 11T^{2} \) |
| 13 | \( 1 - 2.82T + 13T^{2} \) |
| 17 | \( 1 + 4.24T + 17T^{2} \) |
| 23 | \( 1 - 0.585T + 23T^{2} \) |
| 29 | \( 1 - 7.82T + 29T^{2} \) |
| 31 | \( 1 + 0.656T + 31T^{2} \) |
| 37 | \( 1 + 8.24T + 37T^{2} \) |
| 41 | \( 1 - 3.41T + 41T^{2} \) |
| 43 | \( 1 + 2.24T + 43T^{2} \) |
| 47 | \( 1 + 8.82T + 47T^{2} \) |
| 53 | \( 1 + 3.82T + 53T^{2} \) |
| 59 | \( 1 - 8.07T + 59T^{2} \) |
| 61 | \( 1 + 0.828T + 61T^{2} \) |
| 67 | \( 1 + 2T + 67T^{2} \) |
| 71 | \( 1 - 11.8T + 71T^{2} \) |
| 73 | \( 1 - 9.17T + 73T^{2} \) |
| 79 | \( 1 + 5.82T + 79T^{2} \) |
| 83 | \( 1 + 11.1T + 83T^{2} \) |
| 89 | \( 1 + 12.2T + 89T^{2} \) |
| 97 | \( 1 - 8.41T + 97T^{2} \) |
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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
−8.427138323035753333615975731448, −7.12468328331029410010459424238, −6.51414938393302704117793400698, −6.08043676093827681145388475942, −5.18381745550539759255671621656, −4.68075793485890613327447783090, −3.55737725229738493771909416514, −2.69873100326959008744893713363, −2.09191329426343172937934388991, −1.29342874336370697819299494079,
1.29342874336370697819299494079, 2.09191329426343172937934388991, 2.69873100326959008744893713363, 3.55737725229738493771909416514, 4.68075793485890613327447783090, 5.18381745550539759255671621656, 6.08043676093827681145388475942, 6.51414938393302704117793400698, 7.12468328331029410010459424238, 8.427138323035753333615975731448
