L(s) = 1 | − 2-s + 4-s − 8-s − 3·9-s − 4·11-s + 16-s + 4·17-s + 3·18-s − 4·19-s + 4·22-s − 5·25-s − 29-s + 6·31-s − 32-s − 4·34-s − 3·36-s − 2·37-s + 4·38-s + 8·41-s + 4·43-s − 4·44-s − 2·47-s + 5·50-s − 2·53-s + 58-s + 10·59-s + 2·61-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s − 9-s − 1.20·11-s + 1/4·16-s + 0.970·17-s + 0.707·18-s − 0.917·19-s + 0.852·22-s − 25-s − 0.185·29-s + 1.07·31-s − 0.176·32-s − 0.685·34-s − 1/2·36-s − 0.328·37-s + 0.648·38-s + 1.24·41-s + 0.609·43-s − 0.603·44-s − 0.291·47-s + 0.707·50-s − 0.274·53-s + 0.131·58-s + 1.30·59-s + 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2842 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2842 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8608568401\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8608568401\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 7 | \( 1 \) |
| 29 | \( 1 + T \) |
good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−8.623383162880472006797092975494, −8.089257038952552396506260154455, −7.58364168532515406346823242168, −6.51591473524320853471032171322, −5.77279977092827966349271158726, −5.13539331978257867520521506585, −3.88823232548035494580256917114, −2.84304546529151709954683329671, −2.15386751332381269187324279431, −0.60728315265388879974494416609,
0.60728315265388879974494416609, 2.15386751332381269187324279431, 2.84304546529151709954683329671, 3.88823232548035494580256917114, 5.13539331978257867520521506585, 5.77279977092827966349271158726, 6.51591473524320853471032171322, 7.58364168532515406346823242168, 8.089257038952552396506260154455, 8.623383162880472006797092975494