L(s) = 1 | + 2-s − 2·3-s + 4-s − 5-s − 2·6-s + 2·7-s + 8-s + 9-s − 10-s + 5·11-s − 2·12-s − 4·13-s + 2·14-s + 2·15-s + 16-s + 3·17-s + 18-s + 4·19-s − 20-s − 4·21-s + 5·22-s − 3·23-s − 2·24-s − 4·25-s − 4·26-s + 4·27-s + 2·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.15·3-s + 1/2·4-s − 0.447·5-s − 0.816·6-s + 0.755·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s + 1.50·11-s − 0.577·12-s − 1.10·13-s + 0.534·14-s + 0.516·15-s + 1/4·16-s + 0.727·17-s + 0.235·18-s + 0.917·19-s − 0.223·20-s − 0.872·21-s + 1.06·22-s − 0.625·23-s − 0.408·24-s − 4/5·25-s − 0.784·26-s + 0.769·27-s + 0.377·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 195994 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195994 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.572287428\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.572287428\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 43 | \( 1 \) |
| 53 | \( 1 - T \) |
good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 7 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 12 T + p T^{2} \) |
| 79 | \( 1 + 7 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 + 17 T + p T^{2} \) |
| 97 | \( 1 - 3 T + p T^{2} \) |
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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
−12.81859779298408, −12.43404750812067, −11.98722983033169, −11.67080590750521, −11.53053614556068, −11.06893701092253, −10.25801846019776, −9.928485709459493, −9.543441780893920, −8.725413772011728, −8.098364365022710, −7.829322035280497, −7.071206371253789, −6.803254602125945, −6.172564597517093, −5.716772080876430, −5.318030256599567, −4.660154243853382, −4.368970960472128, −3.884937857425336, −3.097246520404091, −2.573197089279582, −1.737506740667935, −1.059383646978325, −0.5994993391837876,
0.5994993391837876, 1.059383646978325, 1.737506740667935, 2.573197089279582, 3.097246520404091, 3.884937857425336, 4.368970960472128, 4.660154243853382, 5.318030256599567, 5.716772080876430, 6.172564597517093, 6.803254602125945, 7.071206371253789, 7.829322035280497, 8.098364365022710, 8.725413772011728, 9.543441780893920, 9.928485709459493, 10.25801846019776, 11.06893701092253, 11.53053614556068, 11.67080590750521, 11.98722983033169, 12.43404750812067, 12.81859779298408