L(s) = 1 | − 2.73·2-s − 3-s + 5.47·4-s − 2.29·5-s + 2.73·6-s − 9.49·8-s + 9-s + 6.26·10-s − 0.920·11-s − 5.47·12-s + 6.70·13-s + 2.29·15-s + 15.0·16-s − 3.80·17-s − 2.73·18-s − 6.56·19-s − 12.5·20-s + 2.51·22-s − 6.79·23-s + 9.49·24-s + 0.244·25-s − 18.3·26-s − 27-s − 4.96·29-s − 6.26·30-s − 1.14·31-s − 22.0·32-s + ⋯ |
L(s) = 1 | − 1.93·2-s − 0.577·3-s + 2.73·4-s − 1.02·5-s + 1.11·6-s − 3.35·8-s + 0.333·9-s + 1.97·10-s − 0.277·11-s − 1.58·12-s + 1.86·13-s + 0.591·15-s + 3.75·16-s − 0.923·17-s − 0.644·18-s − 1.50·19-s − 2.80·20-s + 0.536·22-s − 1.41·23-s + 1.93·24-s + 0.0489·25-s − 3.59·26-s − 0.192·27-s − 0.921·29-s − 1.14·30-s − 0.205·31-s − 3.90·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1709499262\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1709499262\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 41 | \( 1 + T \) |
good | 2 | \( 1 + 2.73T + 2T^{2} \) |
| 5 | \( 1 + 2.29T + 5T^{2} \) |
| 11 | \( 1 + 0.920T + 11T^{2} \) |
| 13 | \( 1 - 6.70T + 13T^{2} \) |
| 17 | \( 1 + 3.80T + 17T^{2} \) |
| 19 | \( 1 + 6.56T + 19T^{2} \) |
| 23 | \( 1 + 6.79T + 23T^{2} \) |
| 29 | \( 1 + 4.96T + 29T^{2} \) |
| 31 | \( 1 + 1.14T + 31T^{2} \) |
| 37 | \( 1 + 10.4T + 37T^{2} \) |
| 43 | \( 1 - 3.89T + 43T^{2} \) |
| 47 | \( 1 - 11.0T + 47T^{2} \) |
| 53 | \( 1 - 7.81T + 53T^{2} \) |
| 59 | \( 1 + 0.168T + 59T^{2} \) |
| 61 | \( 1 + 14.4T + 61T^{2} \) |
| 67 | \( 1 - 1.48T + 67T^{2} \) |
| 71 | \( 1 + 6.79T + 71T^{2} \) |
| 73 | \( 1 + 4.20T + 73T^{2} \) |
| 79 | \( 1 - 4.67T + 79T^{2} \) |
| 83 | \( 1 - 6.73T + 83T^{2} \) |
| 89 | \( 1 - 8.78T + 89T^{2} \) |
| 97 | \( 1 + 1.94T + 97T^{2} \) |
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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
−8.128999680735131787990140710950, −7.62975131693428201299938242345, −6.85189395440563808133127946029, −6.23922915513920231981240674672, −5.69546377157353281073938316955, −4.15517157804988390720637552765, −3.62416156276948513497345228544, −2.28955850841536095768485601332, −1.52357694956425146381458548446, −0.30250418687989259694448616623,
0.30250418687989259694448616623, 1.52357694956425146381458548446, 2.28955850841536095768485601332, 3.62416156276948513497345228544, 4.15517157804988390720637552765, 5.69546377157353281073938316955, 6.23922915513920231981240674672, 6.85189395440563808133127946029, 7.62975131693428201299938242345, 8.128999680735131787990140710950