L(s) = 1 | + 2-s − 3-s − 4-s − 6-s − 7-s − 3·8-s − 2·9-s − 4·11-s + 12-s − 14-s − 16-s + 7·17-s − 2·18-s − 6·19-s + 21-s − 4·22-s + 3·24-s + 5·27-s + 28-s − 29-s + 2·31-s + 5·32-s + 4·33-s + 7·34-s + 2·36-s − 4·37-s − 6·38-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.408·6-s − 0.377·7-s − 1.06·8-s − 2/3·9-s − 1.20·11-s + 0.288·12-s − 0.267·14-s − 1/4·16-s + 1.69·17-s − 0.471·18-s − 1.37·19-s + 0.218·21-s − 0.852·22-s + 0.612·24-s + 0.962·27-s + 0.188·28-s − 0.185·29-s + 0.359·31-s + 0.883·32-s + 0.696·33-s + 1.20·34-s + 1/3·36-s − 0.657·37-s − 0.973·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 - T + p T^{2} \) |
| 3 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 5 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 + 3 T + p T^{2} \) |
| 79 | \( 1 - 9 T + p T^{2} \) |
| 83 | \( 1 + 5 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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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
−15.29486480841818, −14.83621737637754, −14.33137304259446, −13.78769851522391, −13.22378393889962, −12.85596894759802, −12.20649591613081, −11.94950833106387, −11.26774576950370, −10.50615507576961, −10.12887939850003, −9.682978123217032, −8.641907800849525, −8.458116795083976, −7.857623636946168, −6.892612837166375, −6.406649564327144, −5.699511703732277, −5.237185179990350, −5.010605325703916, −4.012067766132286, −3.412213014431586, −2.882465295258751, −2.053845298332588, −0.7253466823521542, 0,
0.7253466823521542, 2.053845298332588, 2.882465295258751, 3.412213014431586, 4.012067766132286, 5.010605325703916, 5.237185179990350, 5.699511703732277, 6.406649564327144, 6.892612837166375, 7.857623636946168, 8.458116795083976, 8.641907800849525, 9.682978123217032, 10.12887939850003, 10.50615507576961, 11.26774576950370, 11.94950833106387, 12.20649591613081, 12.85596894759802, 13.22378393889962, 13.78769851522391, 14.33137304259446, 14.83621737637754, 15.29486480841818