L(s) = 1 | − 2-s − 0.0820·3-s + 4-s − 3.26·5-s + 0.0820·6-s − 0.430·7-s − 8-s − 2.99·9-s + 3.26·10-s + 4.69·11-s − 0.0820·12-s + 7.01·13-s + 0.430·14-s + 0.267·15-s + 16-s + 0.392·17-s + 2.99·18-s + 8.43·19-s − 3.26·20-s + 0.0352·21-s − 4.69·22-s + 23-s + 0.0820·24-s + 5.65·25-s − 7.01·26-s + 0.491·27-s − 0.430·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.0473·3-s + 0.5·4-s − 1.45·5-s + 0.0335·6-s − 0.162·7-s − 0.353·8-s − 0.997·9-s + 1.03·10-s + 1.41·11-s − 0.0236·12-s + 1.94·13-s + 0.114·14-s + 0.0691·15-s + 0.250·16-s + 0.0951·17-s + 0.705·18-s + 1.93·19-s − 0.729·20-s + 0.00770·21-s − 1.00·22-s + 0.208·23-s + 0.0167·24-s + 1.13·25-s − 1.37·26-s + 0.0946·27-s − 0.0812·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.169905513\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.169905513\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 131 | \( 1 + T \) |
good | 3 | \( 1 + 0.0820T + 3T^{2} \) |
| 5 | \( 1 + 3.26T + 5T^{2} \) |
| 7 | \( 1 + 0.430T + 7T^{2} \) |
| 11 | \( 1 - 4.69T + 11T^{2} \) |
| 13 | \( 1 - 7.01T + 13T^{2} \) |
| 17 | \( 1 - 0.392T + 17T^{2} \) |
| 19 | \( 1 - 8.43T + 19T^{2} \) |
| 29 | \( 1 - 0.423T + 29T^{2} \) |
| 31 | \( 1 - 8.04T + 31T^{2} \) |
| 37 | \( 1 + 3.06T + 37T^{2} \) |
| 41 | \( 1 - 6.59T + 41T^{2} \) |
| 43 | \( 1 + 2.90T + 43T^{2} \) |
| 47 | \( 1 + 8.57T + 47T^{2} \) |
| 53 | \( 1 + 10.7T + 53T^{2} \) |
| 59 | \( 1 + 1.19T + 59T^{2} \) |
| 61 | \( 1 - 7.80T + 61T^{2} \) |
| 67 | \( 1 + 4.56T + 67T^{2} \) |
| 71 | \( 1 + 2.37T + 71T^{2} \) |
| 73 | \( 1 - 10.1T + 73T^{2} \) |
| 79 | \( 1 - 2.93T + 79T^{2} \) |
| 83 | \( 1 + 5.21T + 83T^{2} \) |
| 89 | \( 1 - 1.31T + 89T^{2} \) |
| 97 | \( 1 + 8.70T + 97T^{2} \) |
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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
−8.132605128407382899407677425730, −7.62660365415428959748108357094, −6.59381954778422688676714628332, −6.30010451731353182045244356338, −5.28388495785975363953415562597, −4.22885032536442359196253532881, −3.35497087030246200028529540900, −3.18015967696558327568449618135, −1.41164612792037485839577300910, −0.71134213258901092764299978421,
0.71134213258901092764299978421, 1.41164612792037485839577300910, 3.18015967696558327568449618135, 3.35497087030246200028529540900, 4.22885032536442359196253532881, 5.28388495785975363953415562597, 6.30010451731353182045244356338, 6.59381954778422688676714628332, 7.62660365415428959748108357094, 8.132605128407382899407677425730