L(s) = 1 | + 3-s + 9-s + 13-s − 6·17-s − 5·19-s + 6·23-s + 27-s − 6·29-s − 5·31-s + 7·37-s + 39-s + 12·41-s − 43-s + 6·47-s − 6·51-s − 5·57-s + 6·59-s + 2·61-s − 7·67-s + 6·69-s − 12·71-s − 11·73-s + 13·79-s + 81-s − 12·83-s − 6·87-s + 6·89-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 1/3·9-s + 0.277·13-s − 1.45·17-s − 1.14·19-s + 1.25·23-s + 0.192·27-s − 1.11·29-s − 0.898·31-s + 1.15·37-s + 0.160·39-s + 1.87·41-s − 0.152·43-s + 0.875·47-s − 0.840·51-s − 0.662·57-s + 0.781·59-s + 0.256·61-s − 0.855·67-s + 0.722·69-s − 1.42·71-s − 1.28·73-s + 1.46·79-s + 1/9·81-s − 1.31·83-s − 0.643·87-s + 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 - 13 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−14.79003831413870, −14.19382858088299, −13.29647522493350, −13.14254513433307, −12.88321556787922, −12.11432588144928, −11.37694605901413, −10.96290096408654, −10.68555895835646, −9.922010454282218, −9.204581934921366, −8.940890540394544, −8.605564834312902, −7.708956697810334, −7.392556258962761, −6.770019871595133, −6.166767369641604, −5.663075369360187, −4.827524718818926, −4.233347645061875, −3.908072464205827, −3.000659918184536, −2.426121577680317, −1.892329863923324, −0.9898331118794423, 0,
0.9898331118794423, 1.892329863923324, 2.426121577680317, 3.000659918184536, 3.908072464205827, 4.233347645061875, 4.827524718818926, 5.663075369360187, 6.166767369641604, 6.770019871595133, 7.392556258962761, 7.708956697810334, 8.605564834312902, 8.940890540394544, 9.204581934921366, 9.922010454282218, 10.68555895835646, 10.96290096408654, 11.37694605901413, 12.11432588144928, 12.88321556787922, 13.14254513433307, 13.29647522493350, 14.19382858088299, 14.79003831413870