| L(s) = 1 | − 3-s − 3·7-s − 2·9-s + 5·13-s + 3·17-s + 19-s + 3·21-s − 7·23-s + 5·27-s + 29-s + 2·31-s + 2·37-s − 5·39-s − 10·41-s − 6·43-s − 8·47-s + 2·49-s − 3·51-s + 9·53-s − 57-s − 5·59-s − 4·61-s + 6·63-s − 67-s + 7·69-s + 12·71-s + 13·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.13·7-s − 2/3·9-s + 1.38·13-s + 0.727·17-s + 0.229·19-s + 0.654·21-s − 1.45·23-s + 0.962·27-s + 0.185·29-s + 0.359·31-s + 0.328·37-s − 0.800·39-s − 1.56·41-s − 0.914·43-s − 1.16·47-s + 2/7·49-s − 0.420·51-s + 1.23·53-s − 0.132·57-s − 0.650·59-s − 0.512·61-s + 0.755·63-s − 0.122·67-s + 0.842·69-s + 1.42·71-s + 1.52·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 13 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 - 18 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−15.53484319185584, −14.92864136821299, −14.19008376736796, −13.73729572998321, −13.33068708793802, −12.71624278489670, −12.03405597036956, −11.77855605137902, −11.19658289529965, −10.42211098488519, −10.18301471065131, −9.491024935069895, −8.928547561411363, −8.163643117755821, −7.969524509641198, −6.854418704829138, −6.344749579780650, −6.129802447268693, −5.391903590970520, −4.848321167559435, −3.696651783540248, −3.543433730285563, −2.760840317493557, −1.789425288650137, −0.8473268832213925, 0,
0.8473268832213925, 1.789425288650137, 2.760840317493557, 3.543433730285563, 3.696651783540248, 4.848321167559435, 5.391903590970520, 6.129802447268693, 6.344749579780650, 6.854418704829138, 7.969524509641198, 8.163643117755821, 8.928547561411363, 9.491024935069895, 10.18301471065131, 10.42211098488519, 11.19658289529965, 11.77855605137902, 12.03405597036956, 12.71624278489670, 13.33068708793802, 13.73729572998321, 14.19008376736796, 14.92864136821299, 15.53484319185584
