L(s) = 1 | + 2-s − 0.990·3-s + 4-s + 4.23·5-s − 0.990·6-s − 4.18·7-s + 8-s − 2.01·9-s + 4.23·10-s − 4.64·11-s − 0.990·12-s − 2.88·13-s − 4.18·14-s − 4.19·15-s + 16-s + 6.19·17-s − 2.01·18-s + 4.17·19-s + 4.23·20-s + 4.14·21-s − 4.64·22-s + 23-s − 0.990·24-s + 12.9·25-s − 2.88·26-s + 4.97·27-s − 4.18·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.571·3-s + 0.5·4-s + 1.89·5-s − 0.404·6-s − 1.58·7-s + 0.353·8-s − 0.672·9-s + 1.34·10-s − 1.40·11-s − 0.285·12-s − 0.800·13-s − 1.11·14-s − 1.08·15-s + 0.250·16-s + 1.50·17-s − 0.475·18-s + 0.957·19-s + 0.947·20-s + 0.905·21-s − 0.991·22-s + 0.208·23-s − 0.202·24-s + 2.59·25-s − 0.566·26-s + 0.956·27-s − 0.791·28-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
\(L(1)\) |
\(\approx\) |
\(2.577591482\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.577591482\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.990T + 3T^{2} \) |
| 5 | \( 1 - 4.23T + 5T^{2} \) |
| 7 | \( 1 + 4.18T + 7T^{2} \) |
| 11 | \( 1 + 4.64T + 11T^{2} \) |
| 13 | \( 1 + 2.88T + 13T^{2} \) |
| 17 | \( 1 - 6.19T + 17T^{2} \) |
| 19 | \( 1 - 4.17T + 19T^{2} \) |
| 29 | \( 1 + 3.53T + 29T^{2} \) |
| 31 | \( 1 + 6.53T + 31T^{2} \) |
| 37 | \( 1 - 8.77T + 37T^{2} \) |
| 41 | \( 1 - 0.889T + 41T^{2} \) |
| 43 | \( 1 - 9.47T + 43T^{2} \) |
| 47 | \( 1 - 3.37T + 47T^{2} \) |
| 53 | \( 1 + 5.23T + 53T^{2} \) |
| 59 | \( 1 + 14.4T + 59T^{2} \) |
| 61 | \( 1 - 5.45T + 61T^{2} \) |
| 67 | \( 1 - 12.4T + 67T^{2} \) |
| 71 | \( 1 + 8.58T + 71T^{2} \) |
| 73 | \( 1 - 5.07T + 73T^{2} \) |
| 79 | \( 1 + 7.35T + 79T^{2} \) |
| 83 | \( 1 - 2.15T + 83T^{2} \) |
| 89 | \( 1 - 13.5T + 89T^{2} \) |
| 97 | \( 1 - 11.9T + 97T^{2} \) |
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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
−7.75433110105489776861810006864, −7.20133702269399823460645608382, −6.23578203530680747439277781948, −5.79762967626073736719238156579, −5.49058821570721515537974688576, −4.85420755121152180714699053095, −3.30233497317825508516822772658, −2.86733978429785487688579780021, −2.18493560621754381145998913146, −0.74813963978609014955924268332,
0.74813963978609014955924268332, 2.18493560621754381145998913146, 2.86733978429785487688579780021, 3.30233497317825508516822772658, 4.85420755121152180714699053095, 5.49058821570721515537974688576, 5.79762967626073736719238156579, 6.23578203530680747439277781948, 7.20133702269399823460645608382, 7.75433110105489776861810006864