| L(s) = 1 | − 3-s − 3.35·7-s + 9-s − 3.81·11-s + 13-s + 17-s + 4.24·19-s + 3.35·21-s + 6.24·23-s − 27-s − 6.78·29-s − 5.97·31-s + 3.81·33-s + 4.45·37-s − 39-s − 0.621·41-s − 1.54·43-s + 9.14·47-s + 4.24·49-s − 51-s + 7.08·53-s − 4.24·57-s + 0.272·59-s + 4.78·61-s − 3.35·63-s − 1.81·67-s − 6.24·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.26·7-s + 0.333·9-s − 1.14·11-s + 0.277·13-s + 0.242·17-s + 0.974·19-s + 0.731·21-s + 1.30·23-s − 0.192·27-s − 1.26·29-s − 1.07·31-s + 0.663·33-s + 0.733·37-s − 0.160·39-s − 0.0971·41-s − 0.234·43-s + 1.33·47-s + 0.606·49-s − 0.140·51-s + 0.973·53-s − 0.562·57-s + 0.0354·59-s + 0.613·61-s − 0.422·63-s − 0.221·67-s − 0.752·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + 3.35T + 7T^{2} \) |
| 11 | \( 1 + 3.81T + 11T^{2} \) |
| 17 | \( 1 - T + 17T^{2} \) |
| 19 | \( 1 - 4.24T + 19T^{2} \) |
| 23 | \( 1 - 6.24T + 23T^{2} \) |
| 29 | \( 1 + 6.78T + 29T^{2} \) |
| 31 | \( 1 + 5.97T + 31T^{2} \) |
| 37 | \( 1 - 4.45T + 37T^{2} \) |
| 41 | \( 1 + 0.621T + 41T^{2} \) |
| 43 | \( 1 + 1.54T + 43T^{2} \) |
| 47 | \( 1 - 9.14T + 47T^{2} \) |
| 53 | \( 1 - 7.08T + 53T^{2} \) |
| 59 | \( 1 - 0.272T + 59T^{2} \) |
| 61 | \( 1 - 4.78T + 61T^{2} \) |
| 67 | \( 1 + 1.81T + 67T^{2} \) |
| 71 | \( 1 + 9.16T + 71T^{2} \) |
| 73 | \( 1 - 3.32T + 73T^{2} \) |
| 79 | \( 1 + 5.54T + 79T^{2} \) |
| 83 | \( 1 - 6.27T + 83T^{2} \) |
| 89 | \( 1 - 3.78T + 89T^{2} \) |
| 97 | \( 1 - 14.7T + 97T^{2} \) |
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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
−7.27170784948683271495538723535, −6.98372272803536800573515052568, −5.90708745996603064955019558790, −5.59037195374214280462846905440, −4.85806711340994013905219895019, −3.78365514397281470597522610012, −3.18289113047586463420708106926, −2.35329654145435440488806614057, −1.02624521969201650089672914279, 0,
1.02624521969201650089672914279, 2.35329654145435440488806614057, 3.18289113047586463420708106926, 3.78365514397281470597522610012, 4.85806711340994013905219895019, 5.59037195374214280462846905440, 5.90708745996603064955019558790, 6.98372272803536800573515052568, 7.27170784948683271495538723535
