L(s) = 1 | − 2.47·2-s − 1.34·3-s + 4.11·4-s + 2.94·5-s + 3.32·6-s − 2.85·7-s − 5.23·8-s − 1.18·9-s − 7.28·10-s − 11-s − 5.54·12-s − 1.72·13-s + 7.05·14-s − 3.96·15-s + 4.71·16-s − 6.02·17-s + 2.93·18-s − 4.89·19-s + 12.1·20-s + 3.83·21-s + 2.47·22-s + 3.06·23-s + 7.04·24-s + 3.68·25-s + 4.27·26-s + 5.63·27-s − 11.7·28-s + ⋯ |
L(s) = 1 | − 1.74·2-s − 0.777·3-s + 2.05·4-s + 1.31·5-s + 1.35·6-s − 1.07·7-s − 1.85·8-s − 0.396·9-s − 2.30·10-s − 0.301·11-s − 1.59·12-s − 0.479·13-s + 1.88·14-s − 1.02·15-s + 1.17·16-s − 1.46·17-s + 0.692·18-s − 1.12·19-s + 2.71·20-s + 0.837·21-s + 0.527·22-s + 0.639·23-s + 1.43·24-s + 0.736·25-s + 0.838·26-s + 1.08·27-s − 2.21·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1136552488\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1136552488\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 + T \) |
| 547 | \( 1 - T \) |
good | 2 | \( 1 + 2.47T + 2T^{2} \) |
| 3 | \( 1 + 1.34T + 3T^{2} \) |
| 5 | \( 1 - 2.94T + 5T^{2} \) |
| 7 | \( 1 + 2.85T + 7T^{2} \) |
| 13 | \( 1 + 1.72T + 13T^{2} \) |
| 17 | \( 1 + 6.02T + 17T^{2} \) |
| 19 | \( 1 + 4.89T + 19T^{2} \) |
| 23 | \( 1 - 3.06T + 23T^{2} \) |
| 29 | \( 1 + 9.51T + 29T^{2} \) |
| 31 | \( 1 + 2.12T + 31T^{2} \) |
| 37 | \( 1 + 2.03T + 37T^{2} \) |
| 41 | \( 1 + 2.59T + 41T^{2} \) |
| 43 | \( 1 + 10.3T + 43T^{2} \) |
| 47 | \( 1 + 4.16T + 47T^{2} \) |
| 53 | \( 1 - 10.0T + 53T^{2} \) |
| 59 | \( 1 - 4.62T + 59T^{2} \) |
| 61 | \( 1 + 14.6T + 61T^{2} \) |
| 67 | \( 1 + 7.80T + 67T^{2} \) |
| 71 | \( 1 + 16.5T + 71T^{2} \) |
| 73 | \( 1 - 10.1T + 73T^{2} \) |
| 79 | \( 1 - 8.61T + 79T^{2} \) |
| 83 | \( 1 + 8.70T + 83T^{2} \) |
| 89 | \( 1 - 7.51T + 89T^{2} \) |
| 97 | \( 1 + 3.08T + 97T^{2} \) |
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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
−8.381732961662071720613174643096, −7.22155543119067777019426625048, −6.74388654479056455663611114703, −6.19123012678162835595943649401, −5.64300628256851572983345429850, −4.69227669182466420332100686756, −3.19024632050134158407706102289, −2.30310092949870673869280631472, −1.73390558817644495411198964980, −0.22600458316666464249232512515,
0.22600458316666464249232512515, 1.73390558817644495411198964980, 2.30310092949870673869280631472, 3.19024632050134158407706102289, 4.69227669182466420332100686756, 5.64300628256851572983345429850, 6.19123012678162835595943649401, 6.74388654479056455663611114703, 7.22155543119067777019426625048, 8.381732961662071720613174643096