L(s) = 1 | − 2.38·2-s + 1.74·3-s + 3.69·4-s − 3.20·5-s − 4.15·6-s − 0.986·7-s − 4.05·8-s + 0.0355·9-s + 7.64·10-s − 5.86·11-s + 6.44·12-s + 13-s + 2.35·14-s − 5.57·15-s + 2.28·16-s + 6.77·17-s − 0.0848·18-s − 7.90·19-s − 11.8·20-s − 1.71·21-s + 13.9·22-s − 5.15·23-s − 7.06·24-s + 5.24·25-s − 2.38·26-s − 5.16·27-s − 3.65·28-s + ⋯ |
L(s) = 1 | − 1.68·2-s + 1.00·3-s + 1.84·4-s − 1.43·5-s − 1.69·6-s − 0.373·7-s − 1.43·8-s + 0.0118·9-s + 2.41·10-s − 1.76·11-s + 1.86·12-s + 0.277·13-s + 0.629·14-s − 1.43·15-s + 0.571·16-s + 1.64·17-s − 0.0200·18-s − 1.81·19-s − 2.64·20-s − 0.375·21-s + 2.98·22-s − 1.07·23-s − 1.44·24-s + 1.04·25-s − 0.468·26-s − 0.993·27-s − 0.689·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8047 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8047 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0002416687000\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0002416687000\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 13 | \( 1 - T \) |
| 619 | \( 1 + T \) |
good | 2 | \( 1 + 2.38T + 2T^{2} \) |
| 3 | \( 1 - 1.74T + 3T^{2} \) |
| 5 | \( 1 + 3.20T + 5T^{2} \) |
| 7 | \( 1 + 0.986T + 7T^{2} \) |
| 11 | \( 1 + 5.86T + 11T^{2} \) |
| 17 | \( 1 - 6.77T + 17T^{2} \) |
| 19 | \( 1 + 7.90T + 19T^{2} \) |
| 23 | \( 1 + 5.15T + 23T^{2} \) |
| 29 | \( 1 + 2.57T + 29T^{2} \) |
| 31 | \( 1 + 10.3T + 31T^{2} \) |
| 37 | \( 1 + 9.37T + 37T^{2} \) |
| 41 | \( 1 - 3.37T + 41T^{2} \) |
| 43 | \( 1 + 9.97T + 43T^{2} \) |
| 47 | \( 1 + 0.482T + 47T^{2} \) |
| 53 | \( 1 + 6.80T + 53T^{2} \) |
| 59 | \( 1 - 9.59T + 59T^{2} \) |
| 61 | \( 1 - 7.13T + 61T^{2} \) |
| 67 | \( 1 - 0.362T + 67T^{2} \) |
| 71 | \( 1 + 10.2T + 71T^{2} \) |
| 73 | \( 1 - 3.39T + 73T^{2} \) |
| 79 | \( 1 - 1.78T + 79T^{2} \) |
| 83 | \( 1 + 2.94T + 83T^{2} \) |
| 89 | \( 1 - 13.7T + 89T^{2} \) |
| 97 | \( 1 + 9.15T + 97T^{2} \) |
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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
−8.055372447814609050547253529504, −7.59206123038033396340033949968, −7.02045660474134200769138357860, −5.98451511927780824913307901639, −5.08750302862348000784708492326, −3.79175835187416919719715676371, −3.39130034454196118952533622190, −2.47180182724640439461096072126, −1.74144050594678374978313661986, −0.008853462325430701300346327279,
0.008853462325430701300346327279, 1.74144050594678374978313661986, 2.47180182724640439461096072126, 3.39130034454196118952533622190, 3.79175835187416919719715676371, 5.08750302862348000784708492326, 5.98451511927780824913307901639, 7.02045660474134200769138357860, 7.59206123038033396340033949968, 8.055372447814609050547253529504