L(s) = 1 | + 2-s + 1.56·3-s + 4-s − 2.07·5-s + 1.56·6-s + 4.64·7-s + 8-s − 0.555·9-s − 2.07·10-s − 11-s + 1.56·12-s + 6.09·13-s + 4.64·14-s − 3.24·15-s + 16-s + 1.78·17-s − 0.555·18-s + 6.90·19-s − 2.07·20-s + 7.26·21-s − 22-s − 5.48·23-s + 1.56·24-s − 0.689·25-s + 6.09·26-s − 5.55·27-s + 4.64·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.902·3-s + 0.5·4-s − 0.928·5-s + 0.638·6-s + 1.75·7-s + 0.353·8-s − 0.185·9-s − 0.656·10-s − 0.301·11-s + 0.451·12-s + 1.69·13-s + 1.24·14-s − 0.838·15-s + 0.250·16-s + 0.433·17-s − 0.130·18-s + 1.58·19-s − 0.464·20-s + 1.58·21-s − 0.213·22-s − 1.14·23-s + 0.319·24-s − 0.137·25-s + 1.19·26-s − 1.06·27-s + 0.878·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.700045578\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.700045578\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 197 | \( 1 + T \) |
good | 3 | \( 1 - 1.56T + 3T^{2} \) |
| 5 | \( 1 + 2.07T + 5T^{2} \) |
| 7 | \( 1 - 4.64T + 7T^{2} \) |
| 13 | \( 1 - 6.09T + 13T^{2} \) |
| 17 | \( 1 - 1.78T + 17T^{2} \) |
| 19 | \( 1 - 6.90T + 19T^{2} \) |
| 23 | \( 1 + 5.48T + 23T^{2} \) |
| 29 | \( 1 + 4.90T + 29T^{2} \) |
| 31 | \( 1 - 1.11T + 31T^{2} \) |
| 37 | \( 1 - 0.379T + 37T^{2} \) |
| 41 | \( 1 - 2.79T + 41T^{2} \) |
| 43 | \( 1 - 4.01T + 43T^{2} \) |
| 47 | \( 1 - 4.54T + 47T^{2} \) |
| 53 | \( 1 - 7.21T + 53T^{2} \) |
| 59 | \( 1 - 7.85T + 59T^{2} \) |
| 61 | \( 1 + 11.0T + 61T^{2} \) |
| 67 | \( 1 + 3.73T + 67T^{2} \) |
| 71 | \( 1 - 2.38T + 71T^{2} \) |
| 73 | \( 1 + 10.3T + 73T^{2} \) |
| 79 | \( 1 - 14.2T + 79T^{2} \) |
| 83 | \( 1 + 6.38T + 83T^{2} \) |
| 89 | \( 1 + 1.41T + 89T^{2} \) |
| 97 | \( 1 + 9.85T + 97T^{2} \) |
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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
−8.173343861602155990695009638728, −7.77940226925979763368861147110, −7.27209613352403066750754809536, −5.78685840562673075971684811731, −5.52172425192466719296775365276, −4.34011685815381032358762375596, −3.85717285928980855599740515046, −3.14183002297278505540552761916, −2.08467510717804119193039649781, −1.16129431285513574240643744359,
1.16129431285513574240643744359, 2.08467510717804119193039649781, 3.14183002297278505540552761916, 3.85717285928980855599740515046, 4.34011685815381032358762375596, 5.52172425192466719296775365276, 5.78685840562673075971684811731, 7.27209613352403066750754809536, 7.77940226925979763368861147110, 8.173343861602155990695009638728