| L(s) = 1 | + 2-s − 3-s + 4-s + 0.153·5-s − 6-s + 8-s + 9-s + 0.153·10-s + 4.79·11-s − 12-s − 6.39·13-s − 0.153·15-s + 16-s − 4.25·17-s + 18-s − 19-s + 0.153·20-s + 4.79·22-s + 0.168·23-s − 24-s − 4.97·25-s − 6.39·26-s − 27-s + 4.80·29-s − 0.153·30-s + 8.36·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.0684·5-s − 0.408·6-s + 0.353·8-s + 0.333·9-s + 0.0484·10-s + 1.44·11-s − 0.288·12-s − 1.77·13-s − 0.0395·15-s + 0.250·16-s − 1.03·17-s + 0.235·18-s − 0.229·19-s + 0.0342·20-s + 1.02·22-s + 0.0351·23-s − 0.204·24-s − 0.995·25-s − 1.25·26-s − 0.192·27-s + 0.892·29-s − 0.0279·30-s + 1.50·31-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
| \(L(1)\) |
\(\approx\) |
\(2.505710545\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.505710545\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.153T + 5T^{2} \) |
| 11 | \( 1 - 4.79T + 11T^{2} \) |
| 13 | \( 1 + 6.39T + 13T^{2} \) |
| 17 | \( 1 + 4.25T + 17T^{2} \) |
| 23 | \( 1 - 0.168T + 23T^{2} \) |
| 29 | \( 1 - 4.80T + 29T^{2} \) |
| 31 | \( 1 - 8.36T + 31T^{2} \) |
| 37 | \( 1 + 2.84T + 37T^{2} \) |
| 41 | \( 1 - 9.86T + 41T^{2} \) |
| 43 | \( 1 - 9.59T + 43T^{2} \) |
| 47 | \( 1 - 3.03T + 47T^{2} \) |
| 53 | \( 1 + 3.28T + 53T^{2} \) |
| 59 | \( 1 - 14.4T + 59T^{2} \) |
| 61 | \( 1 - 2.72T + 61T^{2} \) |
| 67 | \( 1 - 5.16T + 67T^{2} \) |
| 71 | \( 1 + 1.99T + 71T^{2} \) |
| 73 | \( 1 + 7.84T + 73T^{2} \) |
| 79 | \( 1 - 5.49T + 79T^{2} \) |
| 83 | \( 1 + 3.96T + 83T^{2} \) |
| 89 | \( 1 - 17.4T + 89T^{2} \) |
| 97 | \( 1 + 4.22T + 97T^{2} \) |
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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
−7.948729289956607864437066726537, −7.14054002926987624542201245743, −6.63249101990717785064494524094, −6.02791442103978957105380534643, −5.19691535217086868849753383208, −4.35606550131011319419812310753, −4.10180044451479450692400431083, −2.74393601613416501002927411464, −2.05901699312838220533081007540, −0.77696660038852330870514129752,
0.77696660038852330870514129752, 2.05901699312838220533081007540, 2.74393601613416501002927411464, 4.10180044451479450692400431083, 4.35606550131011319419812310753, 5.19691535217086868849753383208, 6.02791442103978957105380534643, 6.63249101990717785064494524094, 7.14054002926987624542201245743, 7.948729289956607864437066726537
