L(s) = 1 | − 3·3-s + 20·7-s + 9·9-s + 24·11-s − 74·13-s − 54·17-s + 124·19-s − 60·21-s − 120·23-s − 27·27-s − 78·29-s − 200·31-s − 72·33-s + 70·37-s + 222·39-s + 330·41-s + 92·43-s − 24·47-s + 57·49-s + 162·51-s − 450·53-s − 372·57-s − 24·59-s − 322·61-s + 180·63-s − 196·67-s + 360·69-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1.07·7-s + 1/3·9-s + 0.657·11-s − 1.57·13-s − 0.770·17-s + 1.49·19-s − 0.623·21-s − 1.08·23-s − 0.192·27-s − 0.499·29-s − 1.15·31-s − 0.379·33-s + 0.311·37-s + 0.911·39-s + 1.25·41-s + 0.326·43-s − 0.0744·47-s + 0.166·49-s + 0.444·51-s − 1.16·53-s − 0.864·57-s − 0.0529·59-s − 0.675·61-s + 0.359·63-s − 0.357·67-s + 0.628·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + p T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 20 T + p^{3} T^{2} \) |
| 11 | \( 1 - 24 T + p^{3} T^{2} \) |
| 13 | \( 1 + 74 T + p^{3} T^{2} \) |
| 17 | \( 1 + 54 T + p^{3} T^{2} \) |
| 19 | \( 1 - 124 T + p^{3} T^{2} \) |
| 23 | \( 1 + 120 T + p^{3} T^{2} \) |
| 29 | \( 1 + 78 T + p^{3} T^{2} \) |
| 31 | \( 1 + 200 T + p^{3} T^{2} \) |
| 37 | \( 1 - 70 T + p^{3} T^{2} \) |
| 41 | \( 1 - 330 T + p^{3} T^{2} \) |
| 43 | \( 1 - 92 T + p^{3} T^{2} \) |
| 47 | \( 1 + 24 T + p^{3} T^{2} \) |
| 53 | \( 1 + 450 T + p^{3} T^{2} \) |
| 59 | \( 1 + 24 T + p^{3} T^{2} \) |
| 61 | \( 1 + 322 T + p^{3} T^{2} \) |
| 67 | \( 1 + 196 T + p^{3} T^{2} \) |
| 71 | \( 1 - 288 T + p^{3} T^{2} \) |
| 73 | \( 1 - 430 T + p^{3} T^{2} \) |
| 79 | \( 1 - 520 T + p^{3} T^{2} \) |
| 83 | \( 1 - 156 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1026 T + p^{3} T^{2} \) |
| 97 | \( 1 - 286 T + p^{3} T^{2} \) |
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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
−9.218809229194512153942687444368, −7.84563810155871644752362828157, −7.47860198435423264943017229697, −6.44425346285438644134737409155, −5.41305493821322263023790834046, −4.79363682781824030276303563009, −3.88464582951989838467764925781, −2.39271541967963201316391848598, −1.38210918217114272875617648861, 0,
1.38210918217114272875617648861, 2.39271541967963201316391848598, 3.88464582951989838467764925781, 4.79363682781824030276303563009, 5.41305493821322263023790834046, 6.44425346285438644134737409155, 7.47860198435423264943017229697, 7.84563810155871644752362828157, 9.218809229194512153942687444368