L(s) = 1 | + 2.75·2-s + 3-s + 5.59·4-s + 2.75·6-s − 0.393·7-s + 9.92·8-s + 9-s − 0.393·11-s + 5.59·12-s + 2.56·13-s − 1.08·14-s + 16.1·16-s − 2.07·17-s + 2.75·18-s − 0.958·19-s − 0.393·21-s − 1.08·22-s − 6.15·23-s + 9.92·24-s + 7.07·26-s + 27-s − 2.20·28-s − 29-s − 10.1·31-s + 24.6·32-s − 0.393·33-s − 5.72·34-s + ⋯ |
L(s) = 1 | + 1.94·2-s + 0.577·3-s + 2.79·4-s + 1.12·6-s − 0.148·7-s + 3.50·8-s + 0.333·9-s − 0.118·11-s + 1.61·12-s + 0.711·13-s − 0.290·14-s + 4.03·16-s − 0.504·17-s + 0.649·18-s − 0.219·19-s − 0.0859·21-s − 0.231·22-s − 1.28·23-s + 2.02·24-s + 1.38·26-s + 0.192·27-s − 0.416·28-s − 0.185·29-s − 1.82·31-s + 4.36·32-s − 0.0685·33-s − 0.982·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(7.674422592\) |
\(L(\frac12)\) |
\(\approx\) |
\(7.674422592\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 29 | \( 1 + T \) |
good | 2 | \( 1 - 2.75T + 2T^{2} \) |
| 7 | \( 1 + 0.393T + 7T^{2} \) |
| 11 | \( 1 + 0.393T + 11T^{2} \) |
| 13 | \( 1 - 2.56T + 13T^{2} \) |
| 17 | \( 1 + 2.07T + 17T^{2} \) |
| 19 | \( 1 + 0.958T + 19T^{2} \) |
| 23 | \( 1 + 6.15T + 23T^{2} \) |
| 31 | \( 1 + 10.1T + 31T^{2} \) |
| 37 | \( 1 - 7.34T + 37T^{2} \) |
| 41 | \( 1 + 1.65T + 41T^{2} \) |
| 43 | \( 1 + 10.3T + 43T^{2} \) |
| 47 | \( 1 - 11.5T + 47T^{2} \) |
| 53 | \( 1 - 12.3T + 53T^{2} \) |
| 59 | \( 1 + 9.54T + 59T^{2} \) |
| 61 | \( 1 + 6.25T + 61T^{2} \) |
| 67 | \( 1 - 7.42T + 67T^{2} \) |
| 71 | \( 1 - 5.98T + 71T^{2} \) |
| 73 | \( 1 + 3.34T + 73T^{2} \) |
| 79 | \( 1 + 2.06T + 79T^{2} \) |
| 83 | \( 1 + 6.41T + 83T^{2} \) |
| 89 | \( 1 - 15.8T + 89T^{2} \) |
| 97 | \( 1 - 18.4T + 97T^{2} \) |
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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
−8.982992238402728064166143758379, −7.970130272352659534344290205347, −7.30377696681948342502359883164, −6.43124742122294647409996960618, −5.84821367390979261584205216955, −4.95149759972098544138374838681, −4.01380211137965788391604497403, −3.56798579999133028721106480369, −2.51275430120414012084115987021, −1.71802010582544588974539351584,
1.71802010582544588974539351584, 2.51275430120414012084115987021, 3.56798579999133028721106480369, 4.01380211137965788391604497403, 4.95149759972098544138374838681, 5.84821367390979261584205216955, 6.43124742122294647409996960618, 7.30377696681948342502359883164, 7.970130272352659534344290205347, 8.982992238402728064166143758379