| L(s) = 1 | − 0.741·3-s − 2.54·5-s − 0.0722·7-s − 2.44·9-s − 3.91·11-s − 1.31·13-s + 1.88·15-s + 4.63·17-s + 3.41·19-s + 0.0535·21-s − 3.73·23-s + 1.47·25-s + 4.04·27-s − 6.95·29-s − 8.64·31-s + 2.90·33-s + 0.183·35-s − 6.78·37-s + 0.978·39-s + 2.07·41-s − 4.11·43-s + 6.23·45-s − 0.0199·47-s − 6.99·49-s − 3.43·51-s − 5.41·53-s + 9.95·55-s + ⋯ |
| L(s) = 1 | − 0.428·3-s − 1.13·5-s − 0.0273·7-s − 0.816·9-s − 1.17·11-s − 0.366·13-s + 0.487·15-s + 1.12·17-s + 0.783·19-s + 0.0116·21-s − 0.779·23-s + 0.294·25-s + 0.777·27-s − 1.29·29-s − 1.55·31-s + 0.504·33-s + 0.0310·35-s − 1.11·37-s + 0.156·39-s + 0.324·41-s − 0.627·43-s + 0.929·45-s − 0.00291·47-s − 0.999·49-s − 0.480·51-s − 0.744·53-s + 1.34·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4514751931\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4514751931\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 503 | \( 1 - T \) |
| good | 3 | \( 1 + 0.741T + 3T^{2} \) |
| 5 | \( 1 + 2.54T + 5T^{2} \) |
| 7 | \( 1 + 0.0722T + 7T^{2} \) |
| 11 | \( 1 + 3.91T + 11T^{2} \) |
| 13 | \( 1 + 1.31T + 13T^{2} \) |
| 17 | \( 1 - 4.63T + 17T^{2} \) |
| 19 | \( 1 - 3.41T + 19T^{2} \) |
| 23 | \( 1 + 3.73T + 23T^{2} \) |
| 29 | \( 1 + 6.95T + 29T^{2} \) |
| 31 | \( 1 + 8.64T + 31T^{2} \) |
| 37 | \( 1 + 6.78T + 37T^{2} \) |
| 41 | \( 1 - 2.07T + 41T^{2} \) |
| 43 | \( 1 + 4.11T + 43T^{2} \) |
| 47 | \( 1 + 0.0199T + 47T^{2} \) |
| 53 | \( 1 + 5.41T + 53T^{2} \) |
| 59 | \( 1 - 2.00T + 59T^{2} \) |
| 61 | \( 1 - 3.92T + 61T^{2} \) |
| 67 | \( 1 + 11.5T + 67T^{2} \) |
| 71 | \( 1 - 9.21T + 71T^{2} \) |
| 73 | \( 1 - 5.01T + 73T^{2} \) |
| 79 | \( 1 + 6.85T + 79T^{2} \) |
| 83 | \( 1 - 9.70T + 83T^{2} \) |
| 89 | \( 1 - 6.22T + 89T^{2} \) |
| 97 | \( 1 - 12.4T + 97T^{2} \) |
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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
−8.218292311960824621080711044747, −7.67473450045859716504397765476, −7.28161094678952782943553478355, −6.06930152868884862149223735528, −5.39007012672621371736199427296, −4.89271774388273925708088789924, −3.62114260416204378773336999425, −3.23387627522732603239380440944, −1.98430996672221481820539186298, −0.37229323075969599397905295948,
0.37229323075969599397905295948, 1.98430996672221481820539186298, 3.23387627522732603239380440944, 3.62114260416204378773336999425, 4.89271774388273925708088789924, 5.39007012672621371736199427296, 6.06930152868884862149223735528, 7.28161094678952782943553478355, 7.67473450045859716504397765476, 8.218292311960824621080711044747
