L(s) = 1 | − 1.34·2-s − 0.682·3-s − 0.188·4-s − 3.09·5-s + 0.917·6-s + 7-s + 2.94·8-s − 2.53·9-s + 4.16·10-s − 1.56·11-s + 0.128·12-s + 4.46·13-s − 1.34·14-s + 2.10·15-s − 3.58·16-s − 5.43·17-s + 3.41·18-s − 3.67·19-s + 0.584·20-s − 0.682·21-s + 2.10·22-s − 2.00·24-s + 4.56·25-s − 6.01·26-s + 3.77·27-s − 0.188·28-s + 3.50·29-s + ⋯ |
L(s) = 1 | − 0.951·2-s − 0.393·3-s − 0.0944·4-s − 1.38·5-s + 0.374·6-s + 0.377·7-s + 1.04·8-s − 0.844·9-s + 1.31·10-s − 0.472·11-s + 0.0372·12-s + 1.23·13-s − 0.359·14-s + 0.544·15-s − 0.896·16-s − 1.31·17-s + 0.803·18-s − 0.843·19-s + 0.130·20-s − 0.148·21-s + 0.449·22-s − 0.410·24-s + 0.913·25-s − 1.17·26-s + 0.726·27-s − 0.0357·28-s + 0.651·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3703 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3703 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 - T \) |
| 23 | \( 1 \) |
good | 2 | \( 1 + 1.34T + 2T^{2} \) |
| 3 | \( 1 + 0.682T + 3T^{2} \) |
| 5 | \( 1 + 3.09T + 5T^{2} \) |
| 11 | \( 1 + 1.56T + 11T^{2} \) |
| 13 | \( 1 - 4.46T + 13T^{2} \) |
| 17 | \( 1 + 5.43T + 17T^{2} \) |
| 19 | \( 1 + 3.67T + 19T^{2} \) |
| 29 | \( 1 - 3.50T + 29T^{2} \) |
| 31 | \( 1 - 4.86T + 31T^{2} \) |
| 37 | \( 1 - 9.59T + 37T^{2} \) |
| 41 | \( 1 + 3.10T + 41T^{2} \) |
| 43 | \( 1 + 3.05T + 43T^{2} \) |
| 47 | \( 1 - 5.81T + 47T^{2} \) |
| 53 | \( 1 + 9.86T + 53T^{2} \) |
| 59 | \( 1 + 11.8T + 59T^{2} \) |
| 61 | \( 1 - 1.22T + 61T^{2} \) |
| 67 | \( 1 - 11.4T + 67T^{2} \) |
| 71 | \( 1 - 7.37T + 71T^{2} \) |
| 73 | \( 1 - 13.2T + 73T^{2} \) |
| 79 | \( 1 + 14.2T + 79T^{2} \) |
| 83 | \( 1 - 16.3T + 83T^{2} \) |
| 89 | \( 1 - 6.75T + 89T^{2} \) |
| 97 | \( 1 - 8.36T + 97T^{2} \) |
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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
−8.141577303362978613635746079480, −7.914175607795307298329034190421, −6.75770077493630016372436904324, −6.14643370932397172812978594982, −4.88995401547119381963487932126, −4.41954740094993769235791882002, −3.55287963075152836371801293628, −2.33962094028888372957842928312, −0.912330053896723192776426239212, 0,
0.912330053896723192776426239212, 2.33962094028888372957842928312, 3.55287963075152836371801293628, 4.41954740094993769235791882002, 4.88995401547119381963487932126, 6.14643370932397172812978594982, 6.75770077493630016372436904324, 7.914175607795307298329034190421, 8.141577303362978613635746079480