L(s) = 1 | − 2-s − 0.140·3-s + 4-s + 0.140·6-s − 1.14·7-s − 8-s − 2.98·9-s + 3.12·11-s − 0.140·12-s − 2.85·13-s + 1.14·14-s + 16-s + 2.14·17-s + 2.98·18-s + 4.26·19-s + 0.160·21-s − 3.12·22-s − 1.71·23-s + 0.140·24-s + 2.85·26-s + 0.839·27-s − 1.14·28-s − 4.28·29-s + 6.40·31-s − 32-s − 0.438·33-s − 2.14·34-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.0810·3-s + 0.5·4-s + 0.0573·6-s − 0.431·7-s − 0.353·8-s − 0.993·9-s + 0.940·11-s − 0.0405·12-s − 0.793·13-s + 0.304·14-s + 0.250·16-s + 0.519·17-s + 0.702·18-s + 0.977·19-s + 0.0349·21-s − 0.665·22-s − 0.358·23-s + 0.0286·24-s + 0.560·26-s + 0.161·27-s − 0.215·28-s − 0.794·29-s + 1.14·31-s − 0.176·32-s − 0.0762·33-s − 0.367·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 37 | \( 1 + T \) |
good | 3 | \( 1 + 0.140T + 3T^{2} \) |
| 7 | \( 1 + 1.14T + 7T^{2} \) |
| 11 | \( 1 - 3.12T + 11T^{2} \) |
| 13 | \( 1 + 2.85T + 13T^{2} \) |
| 17 | \( 1 - 2.14T + 17T^{2} \) |
| 19 | \( 1 - 4.26T + 19T^{2} \) |
| 23 | \( 1 + 1.71T + 23T^{2} \) |
| 29 | \( 1 + 4.28T + 29T^{2} \) |
| 31 | \( 1 - 6.40T + 31T^{2} \) |
| 41 | \( 1 + 3.24T + 41T^{2} \) |
| 43 | \( 1 + 10.6T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 7.96T + 53T^{2} \) |
| 59 | \( 1 - 2.69T + 59T^{2} \) |
| 61 | \( 1 - 4.57T + 61T^{2} \) |
| 67 | \( 1 + 10.3T + 67T^{2} \) |
| 71 | \( 1 + 7.10T + 71T^{2} \) |
| 73 | \( 1 - 0.261T + 73T^{2} \) |
| 79 | \( 1 + 8.52T + 79T^{2} \) |
| 83 | \( 1 - 1.16T + 83T^{2} \) |
| 89 | \( 1 + 13.6T + 89T^{2} \) |
| 97 | \( 1 + 14.2T + 97T^{2} \) |
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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
−8.876594542691850911164641915233, −8.173182435395257165521158046602, −7.34180286149514231766113099271, −6.54505990865091987913140605835, −5.80446653765215366337044379923, −4.88423769562506654079926811634, −3.53808184843350127664256815147, −2.79500626875052348091352224436, −1.46839150036347553571859932344, 0,
1.46839150036347553571859932344, 2.79500626875052348091352224436, 3.53808184843350127664256815147, 4.88423769562506654079926811634, 5.80446653765215366337044379923, 6.54505990865091987913140605835, 7.34180286149514231766113099271, 8.173182435395257165521158046602, 8.876594542691850911164641915233