L(s) = 1 | + 0.369·2-s − 0.00229·3-s − 1.86·4-s + 2.37·5-s − 0.000846·6-s − 4.42·7-s − 1.42·8-s − 2.99·9-s + 0.876·10-s + 4.37·11-s + 0.00427·12-s − 4.15·13-s − 1.63·14-s − 0.00544·15-s + 3.20·16-s + 2.13·17-s − 1.10·18-s + 1.88·19-s − 4.42·20-s + 0.0101·21-s + 1.61·22-s − 1.32·23-s + 0.00326·24-s + 0.645·25-s − 1.53·26-s + 0.0137·27-s + 8.24·28-s + ⋯ |
L(s) = 1 | + 0.260·2-s − 0.00132·3-s − 0.931·4-s + 1.06·5-s − 0.000345·6-s − 1.67·7-s − 0.504·8-s − 0.999·9-s + 0.277·10-s + 1.32·11-s + 0.00123·12-s − 1.15·13-s − 0.436·14-s − 0.00140·15-s + 0.800·16-s + 0.518·17-s − 0.260·18-s + 0.431·19-s − 0.990·20-s + 0.00221·21-s + 0.344·22-s − 0.276·23-s + 0.000667·24-s + 0.129·25-s − 0.300·26-s + 0.00264·27-s + 1.55·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.331778452\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.331778452\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 43 | \( 1 \) |
good | 2 | \( 1 - 0.369T + 2T^{2} \) |
| 3 | \( 1 + 0.00229T + 3T^{2} \) |
| 5 | \( 1 - 2.37T + 5T^{2} \) |
| 7 | \( 1 + 4.42T + 7T^{2} \) |
| 11 | \( 1 - 4.37T + 11T^{2} \) |
| 13 | \( 1 + 4.15T + 13T^{2} \) |
| 17 | \( 1 - 2.13T + 17T^{2} \) |
| 19 | \( 1 - 1.88T + 19T^{2} \) |
| 23 | \( 1 + 1.32T + 23T^{2} \) |
| 29 | \( 1 - 7.49T + 29T^{2} \) |
| 31 | \( 1 - 2.12T + 31T^{2} \) |
| 37 | \( 1 - 2.46T + 37T^{2} \) |
| 41 | \( 1 - 8.51T + 41T^{2} \) |
| 47 | \( 1 - 8.63T + 47T^{2} \) |
| 53 | \( 1 + 0.533T + 53T^{2} \) |
| 59 | \( 1 - 0.466T + 59T^{2} \) |
| 61 | \( 1 + 8.94T + 61T^{2} \) |
| 67 | \( 1 - 6.26T + 67T^{2} \) |
| 71 | \( 1 + 1.94T + 71T^{2} \) |
| 73 | \( 1 + 1.11T + 73T^{2} \) |
| 79 | \( 1 - 15.8T + 79T^{2} \) |
| 83 | \( 1 + 12.5T + 83T^{2} \) |
| 89 | \( 1 - 4.09T + 89T^{2} \) |
| 97 | \( 1 - 0.625T + 97T^{2} \) |
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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
−9.318989134086038227181103115545, −8.865000501855671411251437280003, −7.66669292279082427782987686673, −6.48164043618461613409758840914, −6.06952653708394804558379047305, −5.34539265047302829196120501435, −4.26376623008518488688602176997, −3.28204966640406405792597794929, −2.55076288764822497742138542844, −0.74337306341142718011422245714,
0.74337306341142718011422245714, 2.55076288764822497742138542844, 3.28204966640406405792597794929, 4.26376623008518488688602176997, 5.34539265047302829196120501435, 6.06952653708394804558379047305, 6.48164043618461613409758840914, 7.66669292279082427782987686673, 8.865000501855671411251437280003, 9.318989134086038227181103115545