L(s) = 1 | − 2.74·2-s − 3-s + 5.52·4-s + 2.74·6-s − 1.18·7-s − 9.65·8-s + 9-s − 3.47·11-s − 5.52·12-s + 5.63·13-s + 3.24·14-s + 15.4·16-s − 4.27·17-s − 2.74·18-s + 8.18·19-s + 1.18·21-s + 9.53·22-s + 6.05·23-s + 9.65·24-s − 15.4·26-s − 27-s − 6.52·28-s + 4.15·29-s − 2.98·31-s − 23.0·32-s + 3.47·33-s + 11.7·34-s + ⋯ |
L(s) = 1 | − 1.93·2-s − 0.577·3-s + 2.76·4-s + 1.11·6-s − 0.446·7-s − 3.41·8-s + 0.333·9-s − 1.04·11-s − 1.59·12-s + 1.56·13-s + 0.866·14-s + 3.85·16-s − 1.03·17-s − 0.646·18-s + 1.87·19-s + 0.258·21-s + 2.03·22-s + 1.26·23-s + 1.97·24-s − 3.02·26-s − 0.192·27-s − 1.23·28-s + 0.772·29-s − 0.535·31-s − 4.06·32-s + 0.605·33-s + 2.01·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5337845326\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5337845326\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 47 | \( 1 + T \) |
good | 2 | \( 1 + 2.74T + 2T^{2} \) |
| 7 | \( 1 + 1.18T + 7T^{2} \) |
| 11 | \( 1 + 3.47T + 11T^{2} \) |
| 13 | \( 1 - 5.63T + 13T^{2} \) |
| 17 | \( 1 + 4.27T + 17T^{2} \) |
| 19 | \( 1 - 8.18T + 19T^{2} \) |
| 23 | \( 1 - 6.05T + 23T^{2} \) |
| 29 | \( 1 - 4.15T + 29T^{2} \) |
| 31 | \( 1 + 2.98T + 31T^{2} \) |
| 37 | \( 1 + 6.90T + 37T^{2} \) |
| 41 | \( 1 - 7.34T + 41T^{2} \) |
| 43 | \( 1 + 2.62T + 43T^{2} \) |
| 53 | \( 1 - 1.77T + 53T^{2} \) |
| 59 | \( 1 - 2.60T + 59T^{2} \) |
| 61 | \( 1 + 9.07T + 61T^{2} \) |
| 67 | \( 1 - 12.8T + 67T^{2} \) |
| 71 | \( 1 - 1.83T + 71T^{2} \) |
| 73 | \( 1 - 1.43T + 73T^{2} \) |
| 79 | \( 1 + 2.99T + 79T^{2} \) |
| 83 | \( 1 - 6.78T + 83T^{2} \) |
| 89 | \( 1 + 11.7T + 89T^{2} \) |
| 97 | \( 1 - 14.9T + 97T^{2} \) |
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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
−8.647174577410032097719030396114, −7.975224343256897775796772581969, −7.17490651676496202132827584625, −6.67040075365204112211720476787, −5.88981059544037076195944788912, −5.11911068053335406901597843636, −3.47726190695928514019588137923, −2.72927133916999448259213871062, −1.50397280225084147668315815064, −0.62364051745302783906321920417,
0.62364051745302783906321920417, 1.50397280225084147668315815064, 2.72927133916999448259213871062, 3.47726190695928514019588137923, 5.11911068053335406901597843636, 5.88981059544037076195944788912, 6.67040075365204112211720476787, 7.17490651676496202132827584625, 7.975224343256897775796772581969, 8.647174577410032097719030396114