L(s) = 1 | − 1.34·2-s − 3-s − 0.196·4-s + 3.48·5-s + 1.34·6-s + 7-s + 2.94·8-s + 9-s − 4.68·10-s − 0.292·11-s + 0.196·12-s − 1.34·14-s − 3.48·15-s − 3.56·16-s − 6.68·17-s − 1.34·18-s − 4.17·19-s − 0.685·20-s − 21-s + 0.393·22-s − 0.510·23-s − 2.94·24-s + 7.17·25-s − 27-s − 0.196·28-s + 6.17·29-s + 4.68·30-s + ⋯ |
L(s) = 1 | − 0.949·2-s − 0.577·3-s − 0.0982·4-s + 1.56·5-s + 0.548·6-s + 0.377·7-s + 1.04·8-s + 0.333·9-s − 1.48·10-s − 0.0882·11-s + 0.0567·12-s − 0.358·14-s − 0.900·15-s − 0.892·16-s − 1.62·17-s − 0.316·18-s − 0.957·19-s − 0.153·20-s − 0.218·21-s + 0.0838·22-s − 0.106·23-s − 0.602·24-s + 1.43·25-s − 0.192·27-s − 0.0371·28-s + 1.14·29-s + 0.855·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.075683153\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.075683153\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + 1.34T + 2T^{2} \) |
| 5 | \( 1 - 3.48T + 5T^{2} \) |
| 11 | \( 1 + 0.292T + 11T^{2} \) |
| 17 | \( 1 + 6.68T + 17T^{2} \) |
| 19 | \( 1 + 4.17T + 19T^{2} \) |
| 23 | \( 1 + 0.510T + 23T^{2} \) |
| 29 | \( 1 - 6.17T + 29T^{2} \) |
| 31 | \( 1 - 5.78T + 31T^{2} \) |
| 37 | \( 1 - 0.978T + 37T^{2} \) |
| 41 | \( 1 + 0.685T + 41T^{2} \) |
| 43 | \( 1 - 1.19T + 43T^{2} \) |
| 47 | \( 1 - 7.19T + 47T^{2} \) |
| 53 | \( 1 + 13.1T + 53T^{2} \) |
| 59 | \( 1 - 8.97T + 59T^{2} \) |
| 61 | \( 1 - 11.3T + 61T^{2} \) |
| 67 | \( 1 - 12.3T + 67T^{2} \) |
| 71 | \( 1 - 7.27T + 71T^{2} \) |
| 73 | \( 1 + 6.76T + 73T^{2} \) |
| 79 | \( 1 + 2.80T + 79T^{2} \) |
| 83 | \( 1 + 6.17T + 83T^{2} \) |
| 89 | \( 1 - 9.88T + 89T^{2} \) |
| 97 | \( 1 - 2.61T + 97T^{2} \) |
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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
−8.606256293770437967711903646100, −8.158102314684297135185890735436, −6.88482338249029138918511470326, −6.51835484969991427695785918959, −5.62569283122265953194289408590, −4.80304128667275839724889850078, −4.24702301754361313654741081306, −2.47021373364816756057608057587, −1.83552203706708239277823911980, −0.74302106059327389029025927391,
0.74302106059327389029025927391, 1.83552203706708239277823911980, 2.47021373364816756057608057587, 4.24702301754361313654741081306, 4.80304128667275839724889850078, 5.62569283122265953194289408590, 6.51835484969991427695785918959, 6.88482338249029138918511470326, 8.158102314684297135185890735436, 8.606256293770437967711903646100